Analysis of international | National education statistics

Wednesday, January 7, 2015

Statistical Analysis of Data

January 07, 2015 No comments:

Slide 1

What is statistics?

Latin “status”---political state—info useful to state (size of population, armed forces etc)

A branch of mathematics concerned with understanding and summarizing collections of numbers

A collection of numerical facts systematically arranged

Slide 2

Descriptive Statistics

Statistics which describe attributes of a sample or population.

Includes measures of central tendency statistics (e.g., mean, median, mode), frequencies, percentages. Minimum, maximum, and range for a data set, variance etc.

Organize and summaries a set of data

Slide 3

Inferential Statistics

Used to make inferences or judgments about a larger population based on the data collected from a small sample drawn from the population.

A key component of inferential statistics is the calculation of statistical significance of a research finding.

1. Involves

· Estimation

· Hypothesis Testing

2. Purpose

· Make Decisions About Population Characteristics

Slide 5

Key Terms

1. Population (Universe)

All Items of Interest

2. Sample

Portion of Population

3. Parameter

Summary Measure about Population

4. Statistic

Summary Measure about Sample

Slide 6

Key Terms

Parameter:

A characteristic of the population. Denoted with Greek letters such as m or

Statistic:

A characteristic of a sample. Denoted with English letters such as X or S.

Sampling Error:

Describes the amount of error that exists between a sample statistic and corresponding population parameter

Slide 7

Slide 8

Some Notations…

Population

All items under consideration by researcher

m = population mean

s = population standard

deviation

N = population size

p = population percentage

Sample

A portion of the population selected for study

x = sample mean

s = sample standard

deviation

n = sample size

p = sample percentage

Slide 9

Descriptive & Inferential Statistics (DS & IS)

DS gather information about a population characteristic (e.g. income) and describe it with a parameter of interest (e.g. mean)

IS uses the parameter to test a hypothesis pertaining to that characteristic. E.g.

Ho: mean income = UD 4,000

H1: mean income < UD 4,000)

The result for hypothesis testing is used to make inference about the characteristic of interest (e.g. Americans ® upper middle income)

Slide 10

Examples of Descriptive and Inferential Statistics

Descriptive Statistics Inferential Statistics

· Graphical * Confidence interval

-Arrange data in tables * Margin of error

-Bar graphs and pie charts * Compare means of two samples

· Numerical - Pre/post scores

-Percentages - t Test

-Averages * Compare means from three samples

-Range - Pre/post and follow-up

· Relationships - ANOVA = analysis of variance

-Correlation coefficient - Levels of Measurement

-Regression analysis

Slide 11

Another characteristic of data, which determines which statistical calculations are meaningful

Nominal: Qualitative data only; categories of names, labels, or qualities; Can’t be ordered (i.e, best to worst) ex: Survey responses of Yes/No

Ordinal: Qualitative/quantitative; can be ordered, but no meaningful subtractions: ex. Grades A, B, C, D, F

Interval: Quantitative only; meaningful subtractions but not ratios, zero is only a position (not “none”) ex: Temperatures

Ratio: Quantitative only, meaningful subtractions and ratios; zero represents “none” ex. Weights of babies

Slide 12

Measures of Central Tendency

“Say you were standing with one foot in the oven and one foot in an ice bucket. According to the average, you should be perfectly comfortable.”

The mode – applies to ratio, interval, ordinal or nominal scales.

The median – applies to ratio, interval and ordinal scales

The mean – applies to ratio and interval scales

Slide 13

Measuring Variability

Range: lowest to highest score

Average Deviation: average distance from the mean

Variance: average squared distance from the mean

Used in later inferential statistics

Standard Deviation: square root of variance

expressed on the same scale as the mean

Slide 13

Parametric statistics

Statistical analysis that attempts to explain the population parameter using a sample

E.g. of statistical parameters: mean, variance, std. dev., R2, t-value, F-ratio, rxy, etc.

It assumes that the distributions of the variables being assessed belong to known parameterized families of probability distributions

Slide 14

Frequencies and Distributions

Frequency-A frequency is the number of times a value is observed in a distribution or the number of times a particular event occurs.

Distribution-When the observed values are arranged in order they are called a rank order distribution or an array. Distributions demonstrate how the frequencies of observations are distributed across a range of values.

The Mode

Defined as the most frequent value (the peak)

· Applies to ratio, interval, ordinal and nominal scales

· Sensitive to sampling error (noise)

· Distributions may be referred to as uni modal, bimodal or multimodal, depending upon the number of peaks

The Median

Defined as the 50th percentile
Applies to ratio, interval and ordinal scales
Can be used for open-ended distributions

The Mean

Applies only to ratio or interval scales

Sensitive to outliers

How to find?

Mean – the average of a group of numbers.

2, 5, 2, 1, 5

Mean = 3

Mean is found by evening out the numbers

2, 5, 2, 1, 5

mean = 3

How to Find the Mean of a Group of Numbers

Step 1 – Add all the numbers.

8, 10, 12, 18, 22, 26

8+10+12+18+22+26 = 96

Step 2 – Divide the sum by the number of addends.

8, 10, 12, 18, 22, 26

8+10+12+18+22+26 = 96

How many addends are there?

Step 2 – Divide the sum by the number of addends.

The mean or average of these numbers is 16.

8, 10, 12, 18, 22, 26

What is the mean of these numbers?

7, 10, 16

26, 33, 41, 52

Median
is in the
Middle

Median – the middle number in a set of ordered numbers.

1, 3, 7, 10, 13

Median = 7

How to Find the Median in a Group of Numbers

Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 27

18, 19, 21, 24, 27

Step 2 – Find the middle number.

21, 18, 24, 19, 27

18, 19, 21, 24, 27

This is your median number.

Step 3 – If there are two middle numbers, find the mean of these two numbers.

18, 19, 21, 25, 27, 28

When to use this measure?

With a non-normal distribution, the median is appropriate

21+ 25 = 46

What is the median of these numbers?

16, 10, 7

7, 10, 16

29, 8, 4, 11, 19

4, 8, 11, 19, 29

31, 7, 2, 12, 14, 19

2, 7, 12, 14, 19, 31 13

12 + 14 = 26 2) 26

Mode
is the most
Popular

Mode – the number that appears most frequently in a set of numbers.

1, 1, 3, 7, 10, 13

Mode = 1

How to Find the Mode in a Group of Numbers

Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 18

18, 18, 19, 21, 24

Step 2 – Find the number that is repeated the most.

21, 18, 24, 19, 18

18, 18, 19, 21, 24

Which number is the mode?

1, 2, 2, 9, 9, 4, 9, 10

1, 2, 2, 4, 9, 9, 9, 10

When to use this measure?

If your data is nominal, you may use the mode and range

Using all three measures provides a more complete picture of the characteristics of your sample set.

Measures of Variability (Dispersion)

Range – applies to ratio, interval, ordinal scales

Semi-interquartile range – applies to ratio, interval, ordinal scales

Variance (standard deviation) – applies to ratio, interval scales

Understanding the variation

The more the data is spread out, the larger the range, variance, SD and SE (Low precision and accuracy)

The more concentrated the data (precise or homogenous), the smaller the range, variance, and standard deviation (high precision and accuracy)

If all the observations are the same, the range, variance, and standard deviation = 0

None of these measures can be negative

Two distant means with little variations are more likely to be significantly different and vice versa

Range

Interval between lowest and highest values

Generally unreliable – changing one value (highest or lowest) can cause large change in range.

Range
is the distance
Between

Range – the difference between the greatest and the least value in a set of numbers.

1, 1, 3, 7, 10, 13

Range = 12

What is the range?

22, 21, 27, 31, 21, 32

21, 21, 22, 27, 31, 32

32 – 21 = 11

How to Find the Range in a Group of Numbers

Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 27

18, 19, 21, 24, 27

Step 2 – Find the lowest and highest numbers.

21, 18, 24, 19, 27

18, 19, 21, 24, 27

Step 3 – Find the difference between these 2 numbers.

18, 19, 21, 24, 27

27 – 18 = 9

The range is 9

Mid-range: Average of the smallest and largest observations

Measure of relative position

Percentiles and Percentile Ranks

Percentile: The score at or below which a given % of scores lie.

Percentile Rank: The percentage of scores at or below a given score

Mid-hinge: The average of the first and third quartiles.

Quartiles:

Observations that divide data into four equal parts.

First Quartile (Q1)

Semi-Interquartile Range

The interquartile range is the interval between the first and third quartile, i.e. between the 25th and 75th percentile.

The semi-inter quartile range is half the interquartile range.

Can be used with open-ended distributions

Unaffected by extreme scores

Example1: the third quartile of students in the Biometry class = ¾ X 36 = 27th item

Example 2: 60th percentile of the class would be 60/100*36 = 21.6 = 22nd item (round off)

Inter-quartile range/deviation

(Mid-spread): Difference between the Third and the First Quartiles, therefore, considers data of central half and ignores the extreme values

Inter-quartile Range = Q3 - Q1

Quartile deviation = (Q3 - Q1)/2

Quartile Deviation

Measures the dispersion of the middle 50% of the distribution

-- rank the data

-- calculate upper and lower quartiles (UQ & LQ)

Number Sample sorted Values

1 25

2 27

3 20

4 23

5 26

6 24

7 19

8 16

9 25

10 18

11 30

12 29

13 32

14 26

15 24

16 21

17 28

18 27

19 20

20 16

21 14

Number Sample Sorted Values Ranked Values

1 25 14

2 27 16

3 20 16

4 23 18

5 26 19

6 24 20

7 19 20

8 16 21

9 25 23

10 18 24

11 30 24

12 29 25

13 32 25

14 26 26

15 24 26

16 21 27

17 28 27

18 27 28

19 20 29

20 16 30

21 14 32

Number Sample sorted Values Ranked Values

1 25 14 LL

2 27 16

3 20 16

4 23 18

5 26 19

6 24 20 LQ or Q1

7 19 20

8 16 21

9 25 23

10 18 24

11 30 24 Md or Q2

12 29 25

13 32 25

14 26 26

15 24 26

16 21 27 UQ or Q3

17 28 27

18 27 28

19 20 29

20 16 30

21 14 32 UL

Variance

Variance is the average of the squared deviations

Closely related to the standard deviation

In order to eliminate negative sign, deviations are squared (squared units e.g. m2)

v = s2

Variance (for a sample)

Steps:

Compute each deviation

Square each deviation

Sum all the squares

Divide by the data size (sample size) minus one: n-1

Example of Variance

Variance = 54/9 = 6

It is a measure of “spread”.

Notice that the larger the deviations (positive or negative) the larger the variance

Population Variance and Standard Deviation

Sample Variance and Standard Deviation

The standard deviation

It is defines as the square root of the variance

Standard deviation (SD):

Positive square root of the variance

SD = + √ S(y-ў)2÷ n

Variance and standard deviation are

useful for probability and hypothesis testing, therefore, is widely used unlike mean deviation

Population parameters and sample statistics

If we are working with samples, the calculation under-estimates the variance and SD which is biased

Therefore, instead of using n, n-1 (degrees of freedom) is used for sample, e.g.

Standard Deviation

example

Example: {4, 7, 6, 3, 8, 6, 7, 4, 5, 3}

Measure of relationship

correlation

Definitions

Correlation is a statistical technique

that is used to measure a relationship

between two variables.

Correlation requires two scores from

each individual (one score from each

of the two variables)

Correlation Coefficients

A correlation coefficient is a statistic that indicates the strength & direction of the relationship b/w 2 variables (or more) for 1 group of participants

Another definition – specifically for Spearman’s rho:

Spearman’s correlation coefficient is a standardized measure of the strength of relationship b/w 2 variables that does not rely on the assumptions of a parametric test (nonparametric data)

Uses Pearson’s correlation coefficient performed on data that have been converted into ranked scores

Distinguishing Characteristics of
Correlation

Correlation procedures involve one

sample containing all pairs of X and Y

scores

Neither variable is called the

independent or dependent variable

Use the individual pair of scores to

create a scatter plot

The scatter plot

Correlation and causality

The fact that there is a relationship

between two variables does not mean that

changes in one variable cause the

changes in the other variable.

A statistical relationship can exist even though one variable does not cause or influence the other.

Correlation research cannot be used to

infer causal relationships between two variables

in the following examples

�� example 1 - correlation coefficient =1

�� example 2 - correlation coefficient =-1

�� example 3 - correlation coefficient =0

�� the correlation coefficient for the parametric case is called the Pearson product moment correlation coefficient (r)

example 1

paired values

A 3 6 9 12 15

B 1 2 3 4 5

�� variable A (income of family) (1000 pounds)

�� variable B (# of cars owned)

�� here is a perfect and positive correlation as one variate increases in precisely the same proportion as the other variate increases

example 2

paired values

A 3 6 9 12 15

B 5 4 3 2 1

�� variable A (income of family) (100

pounds)

�� variable B (# of children)

�� here is a perfect and negative correlation as one variate decreases in precisely the same proportion as the other variate increases

example 3

paired values

A 3 6 9 12 15

B 4 1 3 5 2

variable A (income of family)

variable B (last number of postal code)

here there is almost no correlation because one

variate does not systematically change with the

other. Any association is caused by A and B being

randomly distributed

Correlation coefficients provide a single numerical value to represent the relationship b/w the 2 variables

Correlation coefficients ranges -1 to +1

-1.00 (negative one) a perfect, inverse relationship

+1.00 (positive one) a perfect, direct relationship

0.00 indicates no relationship

Graphic Representations of Correlation

The form of the relationship

In a linear relationship as the X scores increase the Y scores tend to change in one direction only and can be summarised by a straight line

In a non-linear or curvilinear relationship as the X scores change the Y scores do not tend to only increase or only decrease: the Y scores change their direction of change

Computing a correlation

Alternative Formula for the Correlation Coefficient

Computing a Correlation

Non-linearity

2 Types of Correlation Coefficient Tests

1)Pearson r

Full name is “Pearson product-moment correlation coefficient”

r (lower case r & italicized) is the statistic (fact/piece of data obtained from a study of a large quantity of num. data) for this test

2)Spearman’s rho

Full name is “Spearman’s rank-order correlation coefficient”

rho (lower case rho & italicized) is the statistic for this test

Correlation Coefficients & Strength

Strength of relationship is one thing a correlation coefficient test can tell us

Rule of Thumb for strength size (generally)

A correlation coefficient (r or rho)

Value of 0.00 indicates “no relationship”

Values b/w .01 & .24 may be called “weak”

Values b/w .25 & .49 may be called “moderate”

Values b/w .50 & .74 may be called “moderately strong”

Values b/w .75 and .99 may be called “strong”

A value of 1.00 is called “perfect”

Describing strength of relationships with positive or negative values

What is true in the positive is true in the negative

Ex: values b/w .75 & .99 are “very strong” & values b/w -.75 & -.99 are “very strong” though it is an inverse relationship

Correlation Coefficients &Scatterplots

Scatterplots used to visually show trend of data

Tells us

If relationship indicated

Kind of relationship

Outliers – cases differing from general trend

Graph may indicate direction, strength, and/or relationship of two variables

NOTE

It is ESSENTIAL to plot a scatter plot before conducting correlation analysis

If no relationship found in scatter plot,

No need to conduct correlation

When to Use Pearson r

Use Pearson r when:

Looking at relationship b/w 2 scale variables

Interval or ratio measurements

Data not highly skewed

Distribution of scores is approximately symmetrical

Relationship b/w variables is linear

When to Use Spearman’s rho

Use Spearman’s rho when:

One or both variables are ordinal

Ex: college degree, weight, or height given ranking order (i.e. 1 = lightest, 2 = middle, 3 = heaviest)

One or both sets of data are highly skewed

Distributions are not symmetrical

Relationship is not curvilinear

As determined in examination of scatter plot

Spearman Rank Order Correlation

This correlation coefficient is simply the Pearson r calculated on the rankings of the X and Y variables.

Because ranks of N objects are the integers from 1 to N, the sums and sums of squares are known (provided there are no ties).

Spearman Rank Order Correlation

Spearman Rank Order Correlation

Since we know the sum of the scores and the sum of their squares, we automatically know the variance of the integers from 1 to N.

Spearman Rank Order Correlation

Suppose we compute it with N in the denominator instead of

Spearman Rank Order Correlation

Example

Different Scales, Different Measures of Association

Used to describe the linear

relationship between two variables

that are both interval or ratio variables

The symbol for Pearson’s correlation

coefficient is r

The underlying principle of r is that it

compares how consistently each Y

value is paired with each X value in a

linear fashion

The Pearson Correlation formula

degree to which X and Y vary together

r = ---------------------------------------------------

degree to which X and Y vary separately

Co-variability of X and Y

= -----------------------------------------

variability of X and Y separately

∑XY-(∑X)(∑Y)/N

r = -----------------------------------------

√ (∑X*2 –(∑X) *2/N) (∑Y*2 –(∑Y) *2/N)

Degree of freedom=N-2

Sum of Product Deviations

We have used the sum of

squares or SS to measure the amount

of variation or variability for a single

variable

The sum of products or SP provides a

parallel procedure for measuring the

amount of co variation or co variability

between two variables

Definitional Formula

SS =Σ (X- x)(X -x)

or =Σ (Y -y)(Y -y)

Note :

SP =Σ (X -x)(Y- y)

example

X Y XY

1 3 3

2 6 12

4 4 16

5 7 35

ΣX=12

ΣY=20

ΣXY=66

Substituting:

SP = 66 - 12(20)/4

= 66 - 60

= 6

Calculation of Pearson’s
Correlation Coefficient

Pearson’s correlation coefficient is a

ratio comparing the co variability of X

and Y (the numerator) with the

variability of X and Y separately (the

denominator)

SP measures the co variability of X and Y The variability of X and Y is measured by calculating the SS for X and Y scores separately

Pearson correlation coefficient

r = SP / √ SS X SS Y

example

X Y X-X Y-Y (X-X)(Y-Y) (X-X)2 (Y-Y)2

0 1 -6 -1 +6 36 1

10 3 +4 +1 +4 16 1

4 1 -2 -1 +2 4 1

8 2 +2 0 0 4 0

8 3 +2 +1 +2 4 1

SP = 6+4+2+0+2 = 14

SSX = 36+16+4+4+4 = 64

SSY = 1+1+1+0+1

r = SP / √ SS X SS Y

r= 14/√ 64 * 4

14 ÷ 16

= + 0.875

Inferential statistics

Regression

Regression. The best fit line of prediction.

Using a correlation (relationship between variables) to predict one variable from knowing the score on the other variable

Usually a linear regression (finding the best fitting straight line for the data)

Best illustrated in a scatter plot with the regression line also plotted

The scatter plot

In correlation data, it is sometimes useful to

regard one variable as an independent variable and the other as a dependent variable.

In these circumstances, a linear relationship

between two variables X and Y can be

expressed by the equation Y=bX + a

Where Y is the dependent variable, X the

independent variable and b and a are

constants

In the general linear equation the value of

b is called the slope

The slope determines how much the Y

variable will change when X is increased

by one point

The value of a in the general equation is

called the Y-intercept(cutting the graph)

It determines the value of Y when X=0

A regression is a statistical method for studying the relationship between a single dependent variable and one or more independent variables.

In its simplest form a regression specifies a linear relationship between the dependent and independent variables.

Yi = b0 + b1 X1i + b2 X2i + ei

for a given set of observations

In the social sciences, a regression is generally used to represent a causal process.

Y represents the dependent variable

B0 is the intercept (it represents the predicted value of Y if X1 and X2 equal zero.)

X1 and X2 are the independent variables (also called predictors or regressors)

b1 and b2 are called the regression coefficients and provide a measure of the effect of the independent variables on Y (they measure the slope of the line)

e is the stuff not explained by the causal model.

Why use regression?

Regression is used as a way of testing hypotheses about causal relationships.

Specifically, we have hypotheses about whether the independent variables have a positive or a negative effect on the dependent variable.

Just like in our hypothesis tests about variable means, we also would like to be able to judge how confident we are in our inferences.

Standard Error of Estimate

A regression equation, by itself,

allows you to make predictions, but it

does not provide any information

about the accuracy of the predictions

The standard error of estimate gives a

measure of the standard distance

between a regression line and the

actual data points

To calculate the standard error of estimate

Find a sum of squared deviations (SS)

Each deviation will measure the distance

between the actual Y value (data) and the

predicted Ŷ value (regression line)

This sum of squares is commonly called

SSerror

Definition of Standard Error

The standard deviation of the sampling distribution is the standard error. For the mean, it indicates the average distance of the statistic from the parameter.

Example of Height

Raw Data vs. Sampling Distribution

Formula: Standard Error of Mean

To compute the SEM, use:

For our Example:

Standard Error (SE)

It has become popular recently

Researchers often misunderstand and mis- use SE

Variability of observations is SD while variability of 2 or more sample means is SE

Therefore, often called “Standard error of the means” and SD of a set of observations or a population

Covariance

When two variables covary in opposite directions, as smoking and lung capacity do, values tend to be on opposite sides of the group mean. That is, when smoking is above its group mean, lung capacity tends to be below its group mean.

Consequently, by averaging the product of deviation scores, we can obtain a measure of how the variables vary together.

The Sample Covariance

Instead of averaging by dividing by N, we divide by . The resulting formula is

Calculating Covariance

So we obtain

What is Analysis of Variance?

ANOVA is an inferential test designed for use with 3 or more data sets

t-tests are just a form of ANOVA for 2 groups

ANOVA only interested in establishing the existence of a statistical differences, not their direction.

Based upon an F value (R. A. Fisher) which reflects the ratio between systematic and random/error variance…

Procedure for computing 1-way ANOVA for independent samples

Step 1: Complete the table

i.e.

-square each raw score

-total the raw scores for each group

-total the squared scores for each group.

Step 2: Calculate the Grand Total correction factor

GT =

Step 3: Compute total Sum of Squares

SStotal= åX2 - GT

= (åXA2+XB2+XC2) - GT

Step 4: Compute between groups Sum of Squares

SSbet= - GT

= + + - GT

Step 5: Compute within groups Sum of Squares

SSwit= SStotal - SSbet

Step 6: Determine the d.f for each sum of squares

dftotal= (N - 1)

dfbet= (k - 1)

dfwit= (N - k)

Step 7/8: Estimate the Variances & Compute F

Step 9: Consult F distribution table

-d1 is your df for the numerator (i.e. systematic variance)

-d2 is your df for the denominator (i.e. error variance)

Statistical Decision Process

Type I error – rejecting a true null hypothesis. (treatment has an effect when in fact the treatment has no effect)

Alpha level for a hypothesis test is the probability that the test will lead to a Type I error

Alpha and Probability Values

The level of significance that is selected prior to data collection for accepting or rejecting a null hypothesis is called alpha. The level of significance actually obtained after the data have been collected and analyzed is called the probability value, and is indicated by the symbol p.

Inferential Statistics

Level of significance. The second determinant of statistical power is the p value at which the null hypothesis is to be rejected. Statistical power can be increased by lowering the level of significance needed to reject the null hypothesis.

Error Types

Example - Efficacy Test for New drug

Type I error - Concluding that the new drug is better than the standard (HA) when in fact it is no better (H0). Ineffective drug is deemed better.

Type II error - Failing to conclude that the new drug is better (HA) when in fact it is. Effective drug is deemed to be no better.

Non- parametric statistics

Non-parametric methods

So far we assumed that our samples were drawn from normally distributed populations.

techniques that do not make that assumption are called distribution-free or nonparametric tests.

In situations where the normal assumption is appropriate, nonparametric tests are less efficient than traditional parametric methods.

Nonparametric tests frequently make use only of the order of the observations and not the actual values.

Usually do not state hypotheses in terms of a specific parameter

They make vary few assumptions about the population distribution- distribution-free tests.

Suited for data measured in ordinal and nominal scales

Not as sensitive as parametric tests; more likely to fail in detecting a real difference between two treatments

Statistical analysis that attempts to explain the population parameter using a sample without making assumption about the frequency distribution of the assessed variable

In other words, the variable being assessed is distribution-free

Types of nonparametric tests

Chi-square statistic tests for Goodness of Fit (how well the obtained sample proportions fit the population proportions specified by the null hypothesis

Test for independence – tests whether or not there is a relationship between two variables

Non-Parametric Methods

Spearman Rho Rank Order Correlation Coefficient

To calculate the Spearman rho:

Rank the observations on each variable from lowest to highest.

Tied observations are assigned the average of the ranks.

The difference between the ranks on the X and Y variables are summed and squared:

rrho = 1 – [(6åD2)/ n (n2 – 1)

Is there a relationship between the number of cigarettes smoked and severity of illness?

The null and alternative hypotheses are:

HO: There is no relationship between the number of cigarettes smoked and severity of illness

HA: This is a relationship between the number of cigarettes smoked and severity of illness

a = .05

rrho = 1 – [(6åD2)/ n (n2 – 1)]

= 1 – [6(24) / 8(64-1)]

= .71

tcalc = 2.49

tcrit = 2.447, df = 6, p = .05

Since the calculated t is > the critical value of t, we reject the null hypothesis and conclude that there is a statistically significant positive relationship between the number of cigarettes smoked and severity of illness

Use:A non-parametric procedure that we can use to assess the relationship between variables is the Spearman rho.

Goodness of Fit

The chi-square test is a “goodness of fit” test

it answers the question of how well do experimental data fit expectations.

Example

As an example, you count F2 offspring, and get 290 purple and 110 white flowers. This is a total of 400 (290 + 110) offspring.

We expect a 3/4 : 1/4 ratio. We need to calculate the expected numbers (you MUST use the numbers of offspring, NOT the proportion!!!); this is done by multiplying the total offspring by the expected proportions. This we expect 400 * 3/4 = 300 purple, and 400 * 1/4 = 100 white.

Thus, for purple, obs = 290 and exp = 300. For white, obs = 110 and exp = 100.

Chi square formula

Now it's just a matter of plugging into the formula:

2 = (290 - 300)2 / 300 + (110 - 100)2 / 100

= (-10)2 / 300 + (10)2 / 100

= 100 / 300 + 100 / 100

= 0.333 + 1.000

= 1.333.

This is our chi-square value

State H0 H0 : 120

State H1 H1 : ¹

Choose  = 0.05

Choose n n = 100

Choose Test: Z, t, X2 Test (or p Value)

Compute Test Statistic (or compute P value)

Search for Critical Value

Make Statistical Decision rule

Express Decision

Steps in Test of Hypothesis

Determine the appropriate test

Establish the level of significance:α

Formulate the statistical hypothesis

Calculate the test statistic

Determine the degree of freedom

Compare computed test statistic against a tabled/critical value

1. Determine Appropriate Test

Chi Square is used when both variables are measured on a nominal scale.

It can be applied to interval or ratio data that have been categorized into a small number of groups.

It assumes that the observations are randomly sampled from the population.

All observations are independent (an individual can appear only once in a table and there are no overlapping categories).

It does not make any assumptions about the shape of the distribution nor about the homogeneity of variances.

2. Establish Level of Significance

α is a predetermined value

The convention

α = .05

α = .01

α = .001

3. Determine The Hypothesis:
Whether There is an Association or Not

Ho : The two variables are independent

Ha : The two variables are associated

4. Calculating Test Statistics

5. Determine Degrees of Freedom

df = (R-1)(C-1)

6. Compare computed test statistic against a tabled/critical value

The computed value of the Pearson chi- square statistic is compared with the critical value to determine if the computed value is improbable

The critical tabled values are based on sampling distributions of the Pearson chi-square statistic

If calculated c2 is greater than c2 table value, reject Ho

Example

Suppose a researcher is interested in voting preferences on gun control issues.

A questionnaire was developed and sent to a random sample of 90 voters.

The researcher also collects information about the political party membership of the sample of 90 respondents.

Bivariate Frequency Table or Contingency Table

1. Determine Appropriate Test

Party Membership ( 2 levels) and Nominal

Voting Preference ( 3 levels) and Nominal

2. Establish Level of Significance

Alpha of .05

3. Determine The Hypothesis

Ho : There is no difference between D & R in their opinion on gun control issue.

Ha : There is an association between responses to the gun control survey and the party membership in the population.

4. Calculating Test Statistics

5. Determine Degrees of Freedom

df = (R-1)(C-1) =
(2-1)(3-1) = 2

Critical Chi-Square

Critical values for chi-square are found on tables, sorted by degrees of freedom and probability levels. Be sure to use p = 0.05.

If your calculated chi-square value is greater than the critical value from the table, you “reject the null hypothesis”.

If your chi-square value is less than the critical value, you “fail to reject” the null hypothesis (that is, you accept that your genetic theory about the expected ratio is correct).

Chi-Square Table

6. Compare computed test statistic against a tabled/critical value

α = 0.05

df = 2

Critical tabled value = 5.991

Test statistic, 11.03, exceeds critical value

Null hypothesis is rejected

Democrats & Republicans differ significantly in their opinions on gun control issues

Curriculum Development Process

January 07, 2015 No comments:

An effective educational program can only be successful if it is prepared appropriately.

Curriculum is the backbone of educational program which needs proper designing.

The curriculum process consists mostly of five elements or phases.

1 Situational analyses.

2 Formulation of objectives.

3 Selection of content/Scope and sequence.

4 Methods/ Strategies/Actives.

5 Evaluation.

Situational analyses

In order to develop a procedure of Curriculum Organization, one must keep in mind the realities of the situation. The curriculum is the mirror of the traditions, environment and ideas of the concern society. The main purpose of education system to prepare a curriculum to the actual needs of the society. The main functions of curriculum is to preserve and transmit the cultural heritage to next generation. Language, social needs of the society, political and religious situation must be considered while developing curriculum.

The identification of the areas mentioned here help the planner in curriculum development and the selection of objectives, learning material (learning) and appropriate evaluation procedure.

Important aspects of existing situation are as follows:

Ø Geographical condition of the country.

Ø National and international trends.

Ø Cultural and social needs.

Ø System of examinations etc.

Ø Economical conditions.

Ø Age, level and interests of the learners.

Ø Pattern of curriculum to be followed.

Ø Religious condition

Ø Teacher training programmes.

Determination and formulation of learning outcomes-sources of learning outcome

Formulation of Objectives Validation of Educational Objectives:

The work of the curriculum planners to identify such Goals and objectives which fulfill the desires of the society and according the need of national and international demands. In order to reach these goals the curriculum planners clearly state the aims which are related to various fields of studies or subject areas and even related to classroom management and teaching.

Teachers and learner are the first who have the responsibility to achieve the initial targets in class-room. The achievement of these objectives ultimately leads the learner to-wards various categories of life activities, such as:

Continued learning: motivation toward Learning and create thinking skills in learner

Citizenship: Prepare a good citizen who contribute in the economic, local, national, and international level,

Vocational effectiveness: To help the individual on the vocational aspect and made him an economic assets of society

Home responsibilities: Inculcate the sense of responsibility in the Learner to help each other, respect people, and take care of elders and other related experiences, such as food, social and individual activity.

Validation of Educational Objectives

General principles for stating valid objectives for curriculum.

1. Consistency with the ideology of a nation.

2. Consistency and non-contradiction of educational objectives.

3. Behavioristic interpretation.

4. Consistency with social condition.

5. Democratic ideals/relationship.

Fulfillment of basic human needs.

(Reman.M, 2000)

Consistency with the ideology of a Nation

Every nation has certain beliefs and the philosophy of Pakistan is based upon Islam therefore the objectives of our educational system must be regarding the teaching of Islam. The validity and foundation of educational objectives in Pakistan depends upon Quran and Sunnah.

Fulfillment of basic human needs

Man Need the fulfillment of basic needs in life to maintain equilibrium. The objective must help in the attainment of these needs. E.g. Among the basic needs that have been identified are food, clothing, shelter etc...

Consistency and non-contradiction of education objectives:

Educational objectives must not be contradictory to each other. There should be consistency in the objectives at all levels. The objectives must inculcate certain Islamic value and skill in the individual to help him in the real life situation.

Behavioristic interpretation:

Objectives expressed in terms of student behavior are called as behavioral objectives. To create valid, clear and achievable objectives, the curriculum planners have to articulate these objectives for the development and integration of the personalities, economic efficiency, self realization, critical thinking, problem solving ability, understanding of rights.

The objectives that are not put in terms of human behavior are invalid.

Consistency with social conditions:

The objectives of an educational programme are always related to social and culture realities of a nation. In a develop society that is undergoing little or no change, objectives usually are closely related to conditions of that time (updated) , And when a society ideas and progress is slow in adopting new ways of doing things, because of the repaid advancements in the field of science and technology and inability of a society to cope with time.

The fact is that Mass media of communication has grown and Computers are replacing the manpower which is new social realities of life. The old objectives formulated on the basis of old realties now need a revisit and need modification according to the new social and scientific conditions.

Curriculum planners sometimes avoid the new conditions and they feel no need to improve educational objectives with changing time resulting put the development of a society in danger. Hence the curriculum developers have to formulate such type of objectives which are valid with respect to changing needs and the aspects of past culture which they feel essential to preserve as heritage.

Democratic ideals relationship:

Only the democratic ideology fulfills the basic needs of a society and this is the only one that can be used in validating educational objectives in Pakistan.

To apply the democratic values in a society the curriculum planners must relate the objectives and keep in view democratic values and principles.

As the principles of democracy are very difficult so no summarized statement of these can be used successfully in the justification of educational objectives.

But if the objectives are directly related to democratic principles based upon reasoning and critical thinking then they are also called valid.

Selection and organizing of Learning experiences (Content)

Selection of course content for a subject to teach. The subject should encompass all the possible experiences a learner need at that level. All the material which was taught in past and now should be the part of the subject. With the expansion in knowledge new topics emerged with time, they must be incorporated into each subject. With the expansion of knowledge the principle of complete coverage was replace with the principle of representative content because the attainment of total coverage became difficult.

The principles of subject matter selection

1. The course content must be significant in the same field of knowledge:

This is principle for program of studies consisting of specialized courses, with each course being followed by a more advanced course (from simple to complex).

2. The subject matter selected must possess the principle of survival:

The subject matter should have the ability to survive. Those subjects content survived for long which inspire people and benefit mankind, to fulfill the need of the society in spite of continues change in society and culture

3. The subject matter must have the principle of interest:

Keep in view the principle of interest to motivate the learners to learn more.

4. The content or subject matter must be utilized:

(The Principle of Utility)

The Principle of Utility means that the knowledge presented must be helpful in real life situation and the society well benefit from the outcome of the subject content.

5. The course content should contribute to the development of an Islamic society:

The content selected should helpful in the development of character of Muslims. It should inculcate the ideology and values of a true Muslim. It should inculcate the required skills which is beneficial for an Islamic society. There should be no such content which contradict Islamic values

Some common Considerations for the curriculum organizers

• Subject matter should consist of physical and mental activities.

• the subject Content should be helpful in the development of creative abilities in individual

• logical sequence from simple to complex

• the subject Content Help in the attaining of the objectives of the relevant course

• incorporate the best information from all sources in Subject matter

Procedure of Content Selection:

Different procedures for content selection.

The judgmental procedure

The analytical procedure

The consensual procedure

The experimental procedure

The judgmental procedure:

This procedure is all about the vision of the curriculum planner, the success or failure depends upon the curriculum planner. The curriculum planner should have the vision of past, present and future to see the potentialities of all three dimensions. It is very hard that the judgment of curriculum planner will lead to the best selection of subject matter, for that the curriculum planner must be objective in the selection of content and have knowledge for the benefit of others.

The analytical procedure:

The analytical method is the most commonly used method of content selection. It consists of various techniques to collect information regarding subject matter selection:

Conducting interviews
Collecting information through questionnaires
Gathering information through documentary analysis.
Observing the performance of people.

The consensual procedures:

It is the opinion of the people in society who reached to some level of expertise in a field; excellent in the fields of business, industry, agriculture and Experts as teachers, physicians, engineers, and artists etc. these are the people who represent the society.

The experimental procedure

In this method the content is selected after applying different test;

1. Selection of content matter against some standard criterion (say interest).

2. Hypothesis is formulated that the selected content matter meets the criterion (interest)

3. Hypothesis is tested after gathering information from students and teachers and tested with instrument.

Teaching Methods/Strategies

Teaching method play a very important role in the process of curriculum development, this element of curriculum development process help in the attainment of desired objectives. It is the dynamic side of the curriculum because without proper teaching methods one can’t think of achieving the targets goal of education planner, it’s largely depends upon the methods adopted in the classroom for teaching learning process. This process includes:

Teacher’s activities.

Student’s activities.

This is also the work of curriculum planners to suggest proper teaching methods for the suggested curriculum because during planning curriculum they have to keep the methodology in mind, one can’t include such topics in curriculum which are difficult to teach and for such curriculum, and the curriculum planner must have some proper method in mind.

There are various methods of teaching but the curriculum have to keep financial restraint in mind so for that reason they have to suggest such methods which are applicable in classroom . There are methods as lecture, lecture demonstration, problem solving, project, programmed learning etc. For achieving the aims and objectives of the curriculum the curriculum planner must suggest and impose the innovative and active approaches of teaching and learning to initiate the interest of students and teacher as well as the parents.

Bases for Selecting Instructional Methods:

As discuss earlier there are multiple methods of teaching/ instruction, teacher always face the problem of to adopt which method for different lessons. Therefore it is the work of curriculum planner to suggest such methods in advance so the teacher has no problem to search for it.

Guidelines for the selection of teaching / instructional methods:

Achievement of objectives:

For the teacher to adopt a method for teaching learning he must keep in mind the objective of the curriculum, how to get that objective, instructional objectives is the first consideration in planning for teaching. Such a general objective could be achieved through multiple ways, but specific objectives like the student will be able to write an easy on a given topic narrow the choices considerably.

Principles of learning

The teacher should have the knowledge of individual differences, principles and theories of learning while selecting a teaching method for instruction. It would help him in the adaptation of proper method for teaching learning process in the classroom for a larger number of students with different IQs.

Individual learning styles:

Researchers believes that the most effective learning takes place when the teacher used interactive techniques in teaching learning process, the method that suited to the individual student keeping in view the individual differences of the students and impart knowledge to the learner according to his mental ability. “Optimal for one person is not optimal for another”. Cont….

The Rand Corporation Study (1971) supports these findings by stating that “teacher, student, instructional method, and perhaps, other aspects of the educational process interact with each other. Thus a teacher who works well (is effective) with one type of student using one method, he might be ineffective when working with another method. The effectiveness of a teacher, or method, or whatever varies from one situation to another”.

Self-fulfilling processes and educational stratification

The teachers know about the potentialities of the students in classroom and the fact that every child differ from other in the learning process , keeping in view this, the teacher adopt such methods which help all the students in their own way of learning . B.F Skinner stated “we need to find practices which permit all teachers to teach well and under which all students learn as efficiently as their talents permit”

The teacher should try to develop the potentialities of students which they already have and give them opportunities to develop those potentialities to the utmost level.

Facilities, equipment and resources

The teacher know what he has in hand in the form of equipment, audio-visual aids, resources and facilities, therefore he should plan Instructional planning In the light of available resources. It is the ability of a teacher to use minimum resources, equipments of school and achieve maximum outcome, using methods and activities that involve student in a highly active role with the minimum resources.

Accountability

Accountability is also very important factor in the curriculum development process, Teachers, administrators and others who has the responsibility of the education held responsible for the quality of education

The process of accountability means that someone has to justify his work and responsibility to someone else, so it is the circle of accountability where everybody is answerable to other. It involves continued evaluation, review of the people in the process.

It demands results, costs of producing these results. All the stockholders observe and judge the school, the teachers, the administrators and the supervisors whether the students gain certain skills and knowledge for which curriculum are developed, and what methods are they adopting for imparting knowledge of different subjects to students.

Some professionals consider pupil behaviors as the source of methods

According to Shepherd and Regan (1982; p 127) “Methods are content free and not derived from organized subject matter.

It has been argued that methods are derived from an analysis and application of learning theories. The actions, procedures and manipulations of the teacher are not different during instruction or reading or teaching mathematics. Method is like a vehicle, which is empty but can carry a variety of subject matter. This vehicle is created and constructed from generalizations, principles and assumptions”.

Some, professionals feel that every teacher has its own unique method of teaching which represent a form of his personality. Teachers used those methods which they found easy, as It evident from the everyday practice of teachers, even some teacher create their own methods in the classroom to teach, keeping in view the behavior of the students and the situation of that time

Assessment:

Assessment is very important process of the curriculum development. Assessment of student’s academic achievement for the purpose of evaluation of the overall progress of education. Evaluation gives curriculum planner the tools, techniques and processes for defining, gathering and interpreting data relevant to the goals and objectives of the curriculum.

Evaluation helps on all aspects of curriculum planning, administrating and evaluation of the curriculum. it tells, to what extent the curriculum is good and what are the weaknesses of the curriculum and implementing the curriculum, what need to be done to improve the curriculum development process , the administrating process and the teaching methods of the teacher and where system lack the resources .

Statistics for a Psychology Thesis

January 07, 2015 No comments:

Statistics for a Psychology Thesis

The Big Picture: It all starts with a research question. We design or obtain empirical data that might assist in answering a research question. Statistics is a tool for summarising empirical reality and answering questions. Knowing how to link statistical analysis with the research question is a critical skill. One reason that psychology is special is that it attempts to ground its knowledge in empirical reality. We put our ideas to the test. We are taught to be scientist-practitioners.
Staying open minded: There is often a lot of pressure to obtain certain results, support certain hypotheses or test various complex statistical models. My advice: Stuff them all. Be ethical. Stay true to yourself. Let the data speak to you in all its non-conforming brutal honesty. When you analyse data, discard all agendas. If the sample size is too small to say much conclusively, acknowledge this. If the data does not support your hypotheses, accept it and try to understand why. If you have data based on a correlational design, acknowledge that there are many other competing explanations to the particular causal relationship you might be proposing. The whole point of the empirical process is ABSOLUTELY NOT to act as a checkbox for some ill-conceived theory.
Democracy and statistics: Ideologically based positions are common in public debate. Well designed and analysed empirical studies can be powerful in setting out the “facts” that any side of a debate needs to acknowledge. However, empirical research can be biased and hijacked for particular agendas. Having citizens that are able to critically evaluate empirical research and are able to honestly and skilfully conduct and analyse their own research is important for maintaining a healthy democracy. The rhetorical question I ask often is: “Can you create knowledge from empirical observations? Or must you rely on others to digest it for you?”
Statistics as reasoned decision making: Perhaps because of statistics association with mathematics or perhaps because of the way we are taught statistics and associated rules of thumb, it may appear like there is always a right and wrong way to do statistics. In reality, statistics is just like other domains. There are different ways of doing what we do, and the key is to justify our choices based on reasoned decision making. Reasoned decision making involves weighing up the pros and cons of different choices in terms of such factors as the purpose of the analyses, the nature of the data, and recommendations from statistics textbooks and journals. The idea is to explain your reasons in a logical and coherent way just as you would justify any other decision in life.
Null Hypothesis Significance Testing (NHST): a p value indicates the probability of observing results in a sample as or more extreme as those obtained assuming the null hypothesis is true. NHST is a tool for ruling out random sampling as an explanation for the observed relationship. Failing to reject the null hypothesis does not prove the null hypothesis. Statistical significance does not equal practical importance.
A modern orientation to data analysis: Answers to research questions depend on the status of population parameters. Empirical research aims to estimate population parameters (e.g., size of a correlation, size of group differences, etc.). NHST is still relevant. However, confidence intervals around effect sizes and a general orientation of meta-analytic thinking leads to better thinking about research problems, results interpretation and study design than does NHST.
Effect Size: Thinking about effect sizes is a philosophical shift which emphasises thinking about the practical importance of research findings. Effect size measures may be standardised (e.g., cohen’s d, r, odds ratio, etc.) or unstandardised (e.g., difference between group means, unstandardised regression coefficient, etc.). Think about what this means for practitioners using the knowledge. Contextualise the effect size in terms of its statistical definition, prior research in the area, prior research in the broader discipline and only finally using Cohen’s rules of thumb.
Confidence Intervals: Confidence intervals indicate how confident we can be that the population parameter is between given values (e.g., 95% confidence). Confidence intervals focus our attention on population values, which is what theory is all about. They highlight our degree of uncertainty. If the confidence interval includes the null hypothesis value, we know that we do not have a statistically significant result. In this way confidence intervals provide similar information as NHST, but also much more.
Power Analysis: Having an adequate sample size to assess your research question is important. Statistical power is the probability of finding a statistically significant result for a particular parameter in a particular study where the null hypothesis is false. Power increases with larger population effect sizes, larger sample sizes and less stringent alpha. G-Power 3 (just Google G Power 3) is excellent free software for running power analyses.
Accuracy in Parameter Estimation (AIPE): Power analysis is aligned with NHST. AIPE is aligned with confidence intervals around effect sizes and meta-analytic thinking. AIPE attempts to work out the size of the confidence interval we will have for any given sample size and effect size. The aim is to have a sample size that will give us sufficiently small confidence intervals around our obtained effect sizes to draw the conclusions about effect sizes that we want to draw.
Meta Analytic thinking: Meta analytic thinking involves "a) the prospective formulation of study expectations and design by explicitly invoking prior effect size measures and b) the retrospective interpretation of new results, once they are in hand, via explicit, direct comparison with the prior effect sizes in the related literature" (Thompson, 2008, p.28). This approach incorporates the idea that we read the literature in terms of confidence intervals around effect sizes and we design studies with sufficient power to test for the effect size and sufficient potential to refine our estimate of the parameter under study.
Sharing data with the world: Imagine the potential for knowledge advancement if data underlying published articles was readily assessable to be re-analysed. You could learn about data analysis by trying to replicate analyses on data similar to your thesis. You could do meta-analyses using the complete data sets. You could run analyses that the original authors did not report. You could be an active consumer of their results, rather than a passive receiver. Others would be more receptive to your ideas if they could subject your analyses to scrutiny. Such a model fits with the idea of being open minded, distributing knowledge, and emphasising meta-analytic thinking. In many situations concerns about confidentiality, intellectual property, and the data collector’s right to first publish can be overcome. The message: Consider making your data publicly available after you have published it in a journal.
Software: Be aware of the different statistical packages that are available. SPSS is relatively easy to use. “R” (www.r-project.org/) is an open source (i.e., free software) alternative and is worth learning if you want to become a serious data analyst. It has cutting edge features (e.g., polychoric correlations, bootstrapping, reports for psychological tests, meta analysis, multilevel modelling, item analysis, etc.) , amazing potential for automation and customised output, and encourages a better orientation towards running analyses. Results can be fed back into subsequent analyses; graphs and output can be customised to your needs; it forces you to document your analysis process; it generally requires that you know a little more about what you are doing; and it leads to an approach of being responsive to what the data is saying and adjusting analyses accordingly. For an introduction for psychologists, see (personality-project.org/r/r.guide.html).
Learning Statistics: For many people in psychology, statistics is not something done everyday. A strategy is needed to identify and acquire the skills required to analyse your thesis data. Set out a statistical self-development plan possibly in conjunction with a statistical adviser, identifying things such as books and chapters to read, practice exercises to do, formal courses to do, etc. It is important to get practical experience analysing other datasets before you tackle your thesis dataset.
The right books: It is critical to have the right resources. Get a comprehensive multivariate book (Tabachnick & Fiddel – Using Multivariate Statistics or Hair et al – Multivariate Data Analysis). Get a clear, entertaining, insightful and SPSS-focused book (Field – Discovering Statistics Using SPSS). Get an easy to follow SPSS cookbook for doing your thesis (Pallant – SPSS Survival Manual).
Using statistical consultants: be prepared; be clear about your questions; recognise that statistical consultants are there to provide advice about options and that many decisions are intimately tied up with theoretical considerations and should be made by the researcher.
Taking your time: As Wright (2003) so aptly put it: “Conducting data analysis is like drinking a fine wine. It is important to swirl and sniff the wine, to unpack the complex bouquet and to appreciate the experience.” A good dataset often has a lot to say. When we’ve often spent many months designing and collecting data, it is important to give the data the time to speak to us. Often, this will require us to change how we conceptualise the phenomena. Explore the data; produce lots of graphs; consider the individual cases; assess the assumptions; reflect on the statistical models used; reflect on the metrics of the variables used; and value basic descriptive statistics.
Telling a story: The results section should be the most interesting section of a thesis. It should show how your results answer your research question. It should show the reasons for your statistical decisions. It should explain why the statistical output is interesting. You’ve whet the reader’s appetite with the introduction and method, the results section is where you get to convert your empirical observations into a contribution that advances the sum of all human knowledge.

Evaluating the Potential Incorporation of R into Research Methods Education in Psychology

January 07, 2015 No comments:

I was recently completing some professional development activities that required me to write a report on a self-chosen topic related to diversity in student backgrounds. I chose to use the opportunity to reflect on the potential for using R to teach psychology students research methods. I thought I'd share the report in case it interests anyone.

Abstract

Research methods is fundamental to psychology education at university. Recently, open source software called R has become a compelling alternative to the traditionally used proprietary software called SPSS for teaching research methods. However, despite many strong equity and pedagogical arguments for the use of R, there are also many risks associated with its use. This report reviews the literature on the role of technology in research methods university education. It then reviews literature on the diversity of psychology students in terms of motivations, mathematical backgrounds, and career goals. These reviews are then integrated with a pedagogical assessment of the pros and cons of SPSS and R. Finally, recommendations are made regarding how R could be best implemented in psychology research methods teaching.

Introduction

Training in research methods is a fundamental component of university education in psychology. However, for many reasons subjects in research methods are challenging to teach. Students have diverse mathematical, statistical, and computational backgrounds; students often lack motivation as they struggle to see the relevance of statistics. These issues are compounded by undergraduate majors in psychology that typically have several compulsory research methods subjects. Given the competition for entry into fourth year and post-graduate programs, such research methods subjects can be threatening to struggling students.
As with many other universities, research methods in psychology at Deakin University has largely been taught using software called SPSS. This software is typically taught as a menu driven program that is used to analyse data enabling standard data manipulation, analyses, and plotting. While SPSS is relatively user-friendly for standard analyses, there are several problems with teaching students how to use it. In particular, it is very expensive; thus, students can not be assumed to have access to it either from home for doing assignments or in future jobs. In addition, while SPSS makes it easy to perform standard analyses, it is very difficult to alter what SPSS does to perform novel analyses. Thus, for many reasons some lecturers are seeking alternative statistical software for teaching research methods.
While there are many programs for performing statistical analysis, one particularly promising program, known simply as "R", has emerged as a viable alternative to SPSS. R is open source so it is free for students and staff. Thus, students can use R from home when completing assignments, and can use it in any future job. It has a vast array of statistical functionality. Despite these benefits, it does present several challenges to incorporation into psychology. Analyses are typically performed using scripts. It is often less clear how to run certain analyses. The program often assumes a mental model of a statistician rather than an applied researcher.
Thus, the current report had the following aims. The first aim was to evaluate the pros and cons of using R to teaching psychology students research methods. The second aim was to evaluate how best R could be incorporated. In order to achieve these aims, the report is structured into several parts. First, the general literature of software in statistics education is reviewed. A particular focus is placed on diversity in student backgrounds in applied fields. Second, the backgrounds and career goals of psychology students are presented with reference to the literature and practical experience. Third, the pros and cons of using R versus SPSS is presented. Finally, ideas about how best to incorporate R into statistics education are reviewed.

Statistics education and the role of software

There is a substantial literature on statistics education and the role of statistical software in statistics education. Tiskovskaya and Lancaster (2012) provide one review of the challenges in statistics education. Their review is structured around teaching and learning, statistical literacy, and statistics as a profession. Of particular relevance to teaching statistics in psychology they outline several problems and provide relevant references to the statistical literature. With references taken from their paper, these issues include: inability to apply mathematics to real world problems (e.g., Garfield, 1995); mathematics and statistics anxiety and motivation issues in students (e.g., Gal & Ginsburg, 1994); inherent difficulty in students understanding probability and statistics (e.g., Garfield & Ben-Zvi, 2008); problems with background mathematical and statistical knowledge (e.g., Batanero et al 1994); the need to develop statistical literacy which translates into everyday life (e.g., Gal, 2002); and the need to develop assessment tools to evaluate statistical literacy. Tiskovskaya and Lancaster (2012) also reviewed potential statistics teaching reforms. They note that there is a need to provide contextualised practice, foster statistical literacy, and create an active learning environment.
Of particular relevance to the current review of statistical software, Tiskovskaya and Lancaster (2012) discuss the role of technology in statistics education. The importance of technology has increased as computers have become more powerful. This has enabled students to run powerful statistical programs on their computer. Some teachers have used this power to focus instruction on interpretation of statistical results rather than computational mechanics. Chance et al (2007) further note the value of using interactive applets to explore statistical concepts and taking advantage of internet resources in teaching.
Chance et al's (2007) review also summarises several useful suggestions for incorporating technology in statistics education. Moore (1997) notes the importance of balancing using technology as a tool with remembering that the aim is to teach statistics and not the tool per se. Chance et al (2007) notes particularly valuable uses of technology include analysing real datasets, exploring data graphically, and performing simulations. Chance et al (2007) also review statistical software packages for statistics education noting both the advantages and disadvantages of menu-driven applications such as SPSS.
Chance et al (2007) offer several recommendations for incorporating technology into statistical education. First, they highlight the importance of getting students practicing not just performing analyses, but also focusing on interpretation. Second, they recommend that tasks be carefully structured around exploration so that students see the bigger picture and do not get overwhelmed with software implementation issues. Third, collaborative exercises can force students to justify to their fellow students their reasoning. Fourth, they encourage the use of cycles of prediction and testing, which technology can facilitate (e.g., proposing a hypothesis for a simulation and then testing it).
Chance et al (2007) summarise the GAISE report by Franklin and Garfield (2006) on issues to consider when choosing software to teach statistics. These include (a) "ease of data entry, ability to import data in multiple formats, (b) Interactive capabilities, (c) Dynamic linking between data, graphical, and numerical analyses, (d) Ease of use for particular audiences, and (f) Availability to students, portability" (p.19). Franklin and Garfield (2006) also discuss a range of other implementation issues, such as the amount of time to allocate to software exploration, how much the software will be used in the course, and how accessible the software will be outside class. Garfield (1995) suggest that computers should be used to encourage students to explore data using analysis and visualisation tools. Running simulations and exploring resulting properties is also particularly useful. Thus, overall these general considerations regarding statistics education can inform the choice of statistical software. However, the above review also highlights that choice of software is only a small part of the overall unit design process.

Psychology students and the role of statistics

Pathways of psychological studies in Australian universities typically involve completing a three year undergraduate major in psychology, then a fourth year, followed by post-graduate professional or research degrees at masters or doctoral level. As a result of student interest, specialisation, and competition for places, there is a reduction over year levels. From my experience both at Melbourne University and Deakin University, a ball park estimate of the student numbers as a percentage of first year load, would be 40% at second year, 35% at third year, 10% at fourth year, and 3% at postgraduate level. This is from one to two thousand students at first year. Of course these are just rough estimates, but the point is to highlight that there are huge numbers of students getting a basic undergraduate education in psychology; in contrast, the few that go on to fourth year have both a high skill level in psychology also different needs regarding research methods.
Psychology students are taught using the scientist-practitioner model. A big part of science in psychology is research methods and statistics. Students typically complete two or three research methods subjects at undergraduate level, another unit in fourth year, and potentially further units at postgraduate level. The diverse nature of psychology student backgrounds, motivations, and career outcomes can make research methods a difficult subject to design and teach. Psychology undergraduate students also have diverse career goals and outcomes. Many go on to some form of further study. Those that exit at the end of third year have diverse employment outcomes. For example, Borden and Rajecki describe one US sample finding that income was lower than many other majors and that roles included administrative support (17.6%), social worker (12.6%), counsellor (7.6%) along with a diverse range of other jobs. Of those that go on, some will continue with research, but others will go into some form of applied practice.
In terms of research methods in psychology, there are a diverse range of goals. First, research methods is meant to help all students learn to reason about the scientific literature in psychology. Second, for students who continue with psychology research methods should give students the skills to be able to complete a quantitative fourth year and postgraduate thesis. For a subset of students, quantitative skills is part of their marketable skillset that they can take into future employment. Furthermore, for a small group of students who go on to do their PhD and then join academia, research methods skills are fundamental to the continuation of good research and the vitality of the discipline.
In addition to diverse aims are the diverse student backgrounds in psychology. In particular, there are typically no mathematics pre-requisites. By casual observation many students seem motivated to find work in the helping professions, and particularly as clinical psychologists. Many studies have discussed the challenges of teaching statistics to psychology students. For example LaLonde and Gardner (1993) proposed and tested a model of statistics achievement that combined mathematical aptitude and effort with anxiety and motivation as predictors.
Thus, in combination this diversity in background and student goals introduce several challenges when teaching research methods. For some students the main goal is to introduce a moderate degree of statistical literacy. For others, it is essential that they are at least able to analyse their thesis data in a basic way. A final group of advanced students needs skills that will allow them to model their data in a sophisticated way to contribute to the research literature. Thus, there is a tension between presenting ideas in an accessible way for all students versus tailoring the material for advanced students so they can truly excel.
This tension exists in many different aspects of research methods curriculum. Research methods can be taught with varying degrees of mathematical rigour and abstraction. Teaching can emphasise interpreting output or it can emphasise computational processes. It can also vary in the prominence of software versus ideas. In particular the correct choice of statistical software can substantially interact with these issues of balancing rigour with accessibility. In particular, tools like SPSS are more limited than R, but such limits can make standard analyses easier.
Aiken et al (2008) reviewed doctoral education in statistics and found that most surveyed programs were using SAS or SPSS primarily. They described a case study in curricular innovation in terms of novel topics emerging followed by initiatives from substantive researchers. Textbooks and software that make techniques accessible to psychology graduates also facilitate the teaching process. In some respects, as R has become more accessible through usability innovation and as the needs of data analysts have become more advanced, the argument for R has become more compelling.

Whether to use R in psychology research methods

Pros and cons of R

The above review thus provides a background for understanding both statistics education in general and the diversity in the background and goals of psychological students. The following analysis compares and contrasts R and SPSS as software for teaching research methods in psychology. This initial comparison focuses on price, features, usability and other considerations.
In terms of price, an initial benefit of R is that it is free. It is developed under the GNU open-source licence. It is free to the university and free to students. In contrast a student licence to SPSS for a year is around $200; A professional licence is around $2,000; and SPSS charges expensive licencing fees to the university. R would make it easier to get students to complete analyses from home. Requiring students to purchase SPSS creates equity issues and may even encourage some students to engage in software piracy. If as Devlin et al (2008) suggest that essential textbooks create economic hardship, even more expensive statistical software would compound this problem.
In terms of features, SPSS and R both run on Windows, OSX, and Linux. They both support most standard analyses that students may wish to run. However, R has a larger array of contributed packages. SPSS has several features including a data entry tool, a menu-driven GUI, and an output management system for tables and plots that R does not have. R makes it a lot easier to customise analyses, perform reproducible research, and simulations.
In terms of flexibility SPSS and R both have options for performing flexible analyses. However, R makes it a lot easier to gradually introduce customisation by building on standard analyses. It is also flexible in how it can be used because of the open source licence. R is particularly suited to advanced students who can benefit from the easier pathway it provides for growing statistical sophistication.
In terms of usability R and SPSS are quite different. R assumes greater knowledge about statistics. SPSS has an interface that is more familiar to standard Windows-based programs. R is a programming language with a less consistent mental model to standard Windows programs. R has a steeper initial learning curve, but shallower intermediate curve. R encourages students to gradually develop statistical skills. In particular R has several quirks which create difficulties for the novices (e.g., learning details of syntax, escaping spaces in file paths, treating strings as factors versus character variables, etc.). There are also many things that are easy in SPSS that are difficult in R. Some examples include: variable labels and modifying meta data, editing loaded data, browsing loaded data, producing tables of output, viewing and browsing statistical output, generating all the possible bits of output for an analysis, importing data, standard analyses that SPSS already does, and interactive plotting.
R and SPSS can also be compared in terms of existing resources. There are many online resources for both R and SPSS. Psychology-specific R resources exist but are less plentiful than for SPSS. Furthermore, existing psychology supervisors, research methods staff, and tutors are probably more familiar with SPSS which may cause issues when transitioning teaching to R. That said, many supervisors either train their students directly in the software that they want their students to use or they let the student handle details of implementation.

Mental Models

When choosing between SPSS and R it is worth considering the mental models required to use SPSS and R. These mental models both guide what needs to be trained and also may suggest the gap that needs to be closed between students' initial mental models and that which is required by the software.
The SPSS mental model is centred around a dataset. The typical workflow is as follows: (a) import or create data; (b) define meta data; (c) menus guide analysis choice; (d) dialog boxes guide choices within analyses; (e) large amounts of output are produced; (f) instructional material facilitates interpretation of output; (g) output can be copy and pasted into Word or another program for a final report. Custom statistical functions or taking SPSS output and using it as input to subsequent functions is not encouraged for regular users. Thus, overall the system guides the user in the analysis.
In contrast, R requires that the user guides the software. Thus, the R workflow is as follows: (a) Setup raw data in another program; (b) import data where often the user will have multiple datasets, meta datasets, and other data objects (e.g., vectors, tables of output); (c) transform data as required using a range of commands; (d) perform analyses, where command identification may involve a Google search or looking up a book, and understanding arguments in a command can be facilitated by internal documentation and online tools; (e) because the resulting output is minimal, the user often has to ask for specific output using additional commands; (f) much of what is standard in SPSS requires a custom command in R, but also much of which does not exist in SPSS can be readily created by an intermediate user; it is much easier to extract out particular statistical results and use that as input for subsequent functions; (g) while output can be incorporated into Word or Excel, users are encouraged to engage in various workflows that emphasise reproducible research.

Summary

Thus, overall SPSS is well suited to a menu-driven standardised analysis workflow which meets the needs of many psychology students. R is particularly suited to statisticians that need to perform a diverse range of analyses and are more comfortable with computer programming and statistics in general. R requires greater statistical knowledge and it encourages students to have a plan for their analyses. R also requires students to learn more about computing including programming, the command-line, file formats, and advanced file management. The emphasis on commands creates a greater demand on declarative memory which in turn makes R more suited to students who will perform statistical analysis more regularly. However, the flexibility and nature of R means that it can be used in many more contexts than SPSS such as demonstrating statistical ideas through simulation.
Overall, there are clearly pros and cons of both SPSS and R. R is particularly suited to more advanced students. Occasional users may be more productive initially with SPSS. That said, the many students who never go on with any data analysis work, may learn as much or more by using R. It also remains an empirical question to see how different psychology students might handle R. Thus, the remainder of this report focuses on what implementation of how R could be implemented most effectively.

How to use R in psychology research methods

When considering implementation of R in psychology, it is useful to look at existing textbooks and course implementations. When considering textbooks, it is important to note that psychology tends to use a particular subset of statistical analyses. It also often has analysis goals that differ from other fields. For example, there is a greater emphasis on theoretical meaning, effect sizes, complex experimental designs, test reliability, and causal interpretation. While there are many textbooks that teach statistics using R, only recently have books emerged that are specifically designed to teach R to psychology students. The two main books are Andy Field's "Discovering Statistics Using R" and Dan Navarro's "Learning Statistics with R". An alternative model is to take a more generic R textbook or online resource and combine it with a more traditional psychology textbook such as David Howell's "Statistical Methods for Psychology". In particular, there are many user friendly online resources for learning R such as http://www.statmethods.net/ or Venables, Smith and the R Core Team's "An Introduction to R". Whatever textbook option is chosen an important part of learning R involves learning how to get help. Thus, training should include learning how to navigate online learning resources and internet question and answer sites that are very effective in the case of R (e.g., stackoverflow.com).
Dan Navarro (2013) has written a textbook that teaches statistics to psychology students using R. Navarro (2013) presents several argument for using R instead of a different commercial statistics package. These include: (1) the benefits of the software being free and not locking yourself into expensive proprietary software; (2) that R is highly extensible and has many cutting edge statistical techniques; and (3) that R is a programming language and learning to program is a good thing. He also observes that while R has its problems and challenges, overall it provides the best current available option. Thus, overall, his approach is to inspire the student to see the bigger picture about why they are learning R. Navarro then spends two chapters introducing the R programming language. Starting with simple calculations, many basic concepts of variables, assignment, extracting data, and functions are introduced. Then, standard statistical techniques such as ANOVA and regression are presented with R implementations.
Overall, both these textbooks provide insight into how R could be implemented. Teaching with R provides some opportunity to teach statistics in a slightly deeper way. However, various recipes can be provided to perform standard analyses. Teaching R also requires taking a little extra time to teach the language. The menu-driven interface to R called R-Commander also provides a way of introducing R in a more accessible way. The infrastructure provided by R also provides the opportunity to introduce many important topics such as bootstrapping, simulation, power analysis, and customised formulas. Weekly analysis homework not easily possible with SPSS could consolidate R specific skills.
An additional issue of implementation relates to when R should be introduced. Fourth year provides one such opportunity where the students that remain at this level tend to be more capable and have some initial experience in statistics. Fourth year research methods is a very important subject. It is often designed to prepare students to analyse multivariate data. It is also designed to prepare students to be able to analyse data on their own including preliminary analyses, data cleaning, and transformations. R supports all the standard multivariate techniques that are currently taught at fourth year level. These include PCA, factor analysis, logistic regression, DFA, multiple regression, multilevel modelling, CFA, and SEM. R also makes it easier to explore more advanced methods such as bootstrapping and simulations.

Conclusion

Ultimately, it is an empirical question as to whether using R would provide a more effective tools for research methods education in psychology. It may be useful to explore the idea with some low-stakes optional post-graduate training modules in R. Such programs may give a sense of the kinds of practical issues that arise with students when learning to use R. If R is to be rolled out to all of fourth year psychology, this would be a high risk exercise. It would be important to evaluate the student learning outcomes in a broad way. In particular, it would be important to see any effect on analysis performance in fourth year theses.

References

Aiken, L. S., West, S. G., & Millsap, R. E. (2008). Doctoral training in statistics, measurement, and methodology in psychology: Replication and extension of Aiken, West, Sechrest, and Reno's (1990) survey of PhD programs in North America. The American Psychologist, 63(1), 32-50.
Batanero, C., Godino, J., Green, D., and Holmes, P. (1994). Errors and Difficulties in Understanding Introductory Statistical Concepts. International Journal of Mathematical Education in Science and Technology, 25 (4), 527–547.
Borden, V. M., & Rajecki, D. W. (2000). First-year employment outcomes of psychology baccalaureates: Relatedness, preparedness, and prospects.Teaching of Psychology, 27(3), 164-168.
Chance, B., Ben-Zvi, D., Garfield, J., and Medina, E. (2007). The Role of Technology in Improving Student Learning of Statistics. Technology Innovations in Statistics Education, 1(1). url: http://www.escholarship.org/uc/item/8sd2t4rr
Devlin, M., James, R., & Grigg, G. (2008). Studying and working: A national study of student finances and student engagement. Tertiary Education and Management, 14(2), 111-122.
Franklin, C. & Garfield, J. (2006). The GAISE (Guidelines for Assessment and Instruction in Statistics Education) project: Developing statistics education guidelines for pre K-12 and college courses. In G. Burrill (Ed.), 2006 NCTM Yearbook: Thinking and reasoning with data and chance. Reston, VA: National Council of Teachers of Mathematics.
Gal, I. (2002). Adults' Statistical Literacy: Meanings, Components, Responsibilities. With Discussion. International Statistical Review, 70(1), 1-51.
Gal, I. & Ginsburg, L. (1994). The Role of Beliefs and Attitudes in Learning Statistics: Towards an Assessment Framework. Journal of Statistics Education, 2(2). url: http://www.amstat.org/publications/jse/v2n2/gal.html
Garfield, J. (1995). How Students Learn Statistics. International Statistical Review, 63(1), 25-34.
Garfield, J. and Ben-Zvi, D. (2008). Developing Students' Statistical Reasoning: Connecting Research and Teaching Practice, Springer.
Lalonde, R. N., & Gardner, R. C. (1993). Statistics as a second language? A model for predicting performance in psychology students. Canadian Journal of Behavioural Science, 25, 108-125.
Moore, D.S. (1997). New pedagogy and new content: the case of statistics. International Statistical Review, 635, 123-165.
Navarro, D. (2013). Learning statistics with R: A tutorial for psychology students and other beginners (Version 0.3) url: http://ua.edu.au/ccs/teaching/lsr
Tishkovskaya, S., & Lancaster, G. (2012). Statistical education in the 21st century: a review of challenges, teaching innovations and strategies for reform. Journal of Statistics Education, 20(2), 1-55.
Wiberg, M. (2009). Teaching statistics in integration with psychology. Journal of Statistics Education, 17(1), 1-16.

Analysis of international | National education statistics

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Wednesday, January 7, 2015

Statistical Analysis of Data

Curriculum Development Process

Statistics for a Psychology Thesis

Statistics for a Psychology Thesis

Evaluating the Potential Incorporation of R into Research Methods Education in Psychology

Abstract

Introduction

Statistics education and the role of software

Psychology students and the role of statistics

Whether to use R in psychology research methods

Pros and cons of R

Mental Models

Summary

How to use R in psychology research methods

Conclusion

References

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