4PAHPRM2 Research Methods 2 Data Concepts & Data Analysis
BSc Psychology Year 1
TABLE OF CONTENTS
Chi-Square Tests................................................................................................. 3
Chi-Square Tests ..........................................................................................................................3
The Chi-Square (𝛘𝟐) Statistic ........................................................................................................3
The Chi-Square Goodness-of-Fit Test ........................................................................... 3
Degrees of Freedom .....................................................................................................................3
p-Value ........................................................................................................................................3
The Critical Value .........................................................................................................................3
Equal Expected Frequencies .........................................................................................................4
Unequal Expected Frequencies ....................................................................................................4
Testing for Normally Distributed Data ............................................................................................5
Chi Square Test For Contingency Tables ...................................................................... 5
Calculating Expected Frequencies ................................................................................................6
Calculating the df for a Chi-Square Test for Contingency Tables .....................................................6
Chi-Square for Larger Contingency Tables .....................................................................................6
Phi (ɸ) Correlation ........................................................................................................................7
Column Percentages ....................................................................................................................7
Fisher’s Exact Test ...................................................................................................... 8
McNemar Chi-Square Test........................................................................................... 8
Reporting Chi-Square Tests ......................................................................................... 8
Correlation and Regression ................................................................................. 9
A Regression Line........................................................................................................ 9
The Regression Equation ..............................................................................................................9
The Criterion of Least Squares .................................................................................................... 10
Simple Linear Regression .......................................................................................... 11
When to Use Regression ............................................................................................................. 11
Variables Entered/Removed ....................................................................................................... 12
Model Summary ......................................................................................................................... 12
ANOVA ...................................................................................................................................... 12
Coefficients ............................................................................................................................... 13
Reporting Simple Linear Regression ............................................................................................ 13
Null and Alternate Hypothesis for regression ............................................................. 13
Binary Predictor Variables ......................................................................................... 14
Mean Differences ....................................................................................................................... 14
Transformations ......................................................................................................................... 15
Parametric Tests ............................................................................................... 15
t-tests ...................................................................................................................... 15
Statistical Assumptions for Using t-tests ..................................................................................... 15
Critical Value for t....................................................................................................................... 16
Null Hypothesis Significance Testing (NHST) ............................................................................... 17
z-tests ...................................................................................................................... 17
Non-Parametric Tests ....................................................................................... 18
1
,4PAHPRM2 Research Methods 2 Data Concepts & Data Analysis
BSc Psychology Year 1
Mann-Whitney U Test ................................................................................................ 18
When to Use a Mann-Whitney U Test ........................................................................................... 18
p-value ...................................................................................................................................... 18
Reporting a Mann-Whitney U Test ............................................................................................... 18
Wilcoxon’s matched-pairs signed-ranks test ............................................................. 19
When to Use a Wilcoxon’s Matched-Pairs Signed-Ranks Test ....................................................... 20
p-value ...................................................................................................................................... 20
Reporting a Wilcoxon’s Matched-Pairs Signed-Ranks Test ............................................................ 20
Confidence Intervals ......................................................................................... 21
A Point Estimate ......................................................................................................................... 21
An Interval Estimate .................................................................................................. 21
Confidence Limits ...................................................................................................................... 21
Levels of Confidence Intervals .................................................................................................... 21
Interpreting Confidence Intervals ................................................................................................ 22
Bootstrapping ........................................................................................................... 22
Power and EFFECT size...................................................................................... 22
Statistical Power....................................................................................................... 22
What Affects Power .................................................................................................................... 23
Power Calculation ...................................................................................................................... 23
Effect sizes ............................................................................................................... 23
Standardised Measures of Effect Size .......................................................................................... 23
Comparing Effect Sizes............................................................................................................... 23
Meta-Analysis ........................................................................................................... 24
The File Drawer Problem ............................................................................................................. 24
Effect Sizes in Meta-Analyses...................................................................................................... 24
Confidence Intervals .................................................................................................................. 26
Psychometrics & Questionnaire Data ................................................................. 27
Internal consistency ................................................................................................. 27
Examining Correlation ................................................................................................................ 27
Cronbach’s Alpha (𝜶) ................................................................................................................. 27
Reverse Coding ......................................................................................................... 28
Meta-Analysis & Open Science .......................................................................... 29
Other SPSS commands........................................................................................ 29
Transpose Data in Excel ............................................................................................ 29
References ........................................................................................................ 30
2
, 4PAHPRM2 Research Methods 2 Data Concepts & Data Analysis
BSc Psychology Year 1
CHI-SQUARE TESTS
Chi-Square Tests
• The chi-square test is a non-parametric inferential statistical test for analysing categorical data
• The test examines the discrepancy between expected frequencies and observed frequencies
o We are interested in drawing conclusions about the distribution of scores for a single
categorical variable
Deciding Which Chi-Square Test to Use
Test Name When to Use Design Variables Involved
You have one categorical variable and One- 1 categorical (with ≥2
Chi-square
want to see if its distribution matches sample categories)
Goodness of Fit
expected values
Chi-square Test You want to test for an association Between- 2 categorical (with any
for Contingency between two categorical variables subjects number of levels)
Tables
You have paired categorical data Within- 1 dichotomous variable
McNemar’s Test
subjects measured twice
The Chi-Square (𝛘𝟐 ) Statistic
• A single value, expressing magnitude of the difference between the observed and expected values
• Usually, the bigger the difference between observed and expected values, the bigger χ"
($%&)!
χ" = Σ &
where O is the observed frequencies, and E is the expected frequencies
THE CHI-SQUARE GOODNESS-OF-FIT TEST
Degrees of Freedom
• The degrees of freedom for the chi-square goodness-of-fit test are always one less than the
number of categories
p-Value
• The probability of obtaining this value of χ" or larger, when sampling from a population that generates
our expected frequencies
• The p-value depends on the size of χ" and the degrees of freedom
o The larger χ" , the smaller the p-value
o The larger the df, the smaller the p-value
The Critical Value
• Whenever there is an χ" value above 3.84, the p-value is < .05
o Therefore, the observed values significantly differ from the expected values, so there is a
significant difference
3
BSc Psychology Year 1
TABLE OF CONTENTS
Chi-Square Tests................................................................................................. 3
Chi-Square Tests ..........................................................................................................................3
The Chi-Square (𝛘𝟐) Statistic ........................................................................................................3
The Chi-Square Goodness-of-Fit Test ........................................................................... 3
Degrees of Freedom .....................................................................................................................3
p-Value ........................................................................................................................................3
The Critical Value .........................................................................................................................3
Equal Expected Frequencies .........................................................................................................4
Unequal Expected Frequencies ....................................................................................................4
Testing for Normally Distributed Data ............................................................................................5
Chi Square Test For Contingency Tables ...................................................................... 5
Calculating Expected Frequencies ................................................................................................6
Calculating the df for a Chi-Square Test for Contingency Tables .....................................................6
Chi-Square for Larger Contingency Tables .....................................................................................6
Phi (ɸ) Correlation ........................................................................................................................7
Column Percentages ....................................................................................................................7
Fisher’s Exact Test ...................................................................................................... 8
McNemar Chi-Square Test........................................................................................... 8
Reporting Chi-Square Tests ......................................................................................... 8
Correlation and Regression ................................................................................. 9
A Regression Line........................................................................................................ 9
The Regression Equation ..............................................................................................................9
The Criterion of Least Squares .................................................................................................... 10
Simple Linear Regression .......................................................................................... 11
When to Use Regression ............................................................................................................. 11
Variables Entered/Removed ....................................................................................................... 12
Model Summary ......................................................................................................................... 12
ANOVA ...................................................................................................................................... 12
Coefficients ............................................................................................................................... 13
Reporting Simple Linear Regression ............................................................................................ 13
Null and Alternate Hypothesis for regression ............................................................. 13
Binary Predictor Variables ......................................................................................... 14
Mean Differences ....................................................................................................................... 14
Transformations ......................................................................................................................... 15
Parametric Tests ............................................................................................... 15
t-tests ...................................................................................................................... 15
Statistical Assumptions for Using t-tests ..................................................................................... 15
Critical Value for t....................................................................................................................... 16
Null Hypothesis Significance Testing (NHST) ............................................................................... 17
z-tests ...................................................................................................................... 17
Non-Parametric Tests ....................................................................................... 18
1
,4PAHPRM2 Research Methods 2 Data Concepts & Data Analysis
BSc Psychology Year 1
Mann-Whitney U Test ................................................................................................ 18
When to Use a Mann-Whitney U Test ........................................................................................... 18
p-value ...................................................................................................................................... 18
Reporting a Mann-Whitney U Test ............................................................................................... 18
Wilcoxon’s matched-pairs signed-ranks test ............................................................. 19
When to Use a Wilcoxon’s Matched-Pairs Signed-Ranks Test ....................................................... 20
p-value ...................................................................................................................................... 20
Reporting a Wilcoxon’s Matched-Pairs Signed-Ranks Test ............................................................ 20
Confidence Intervals ......................................................................................... 21
A Point Estimate ......................................................................................................................... 21
An Interval Estimate .................................................................................................. 21
Confidence Limits ...................................................................................................................... 21
Levels of Confidence Intervals .................................................................................................... 21
Interpreting Confidence Intervals ................................................................................................ 22
Bootstrapping ........................................................................................................... 22
Power and EFFECT size...................................................................................... 22
Statistical Power....................................................................................................... 22
What Affects Power .................................................................................................................... 23
Power Calculation ...................................................................................................................... 23
Effect sizes ............................................................................................................... 23
Standardised Measures of Effect Size .......................................................................................... 23
Comparing Effect Sizes............................................................................................................... 23
Meta-Analysis ........................................................................................................... 24
The File Drawer Problem ............................................................................................................. 24
Effect Sizes in Meta-Analyses...................................................................................................... 24
Confidence Intervals .................................................................................................................. 26
Psychometrics & Questionnaire Data ................................................................. 27
Internal consistency ................................................................................................. 27
Examining Correlation ................................................................................................................ 27
Cronbach’s Alpha (𝜶) ................................................................................................................. 27
Reverse Coding ......................................................................................................... 28
Meta-Analysis & Open Science .......................................................................... 29
Other SPSS commands........................................................................................ 29
Transpose Data in Excel ............................................................................................ 29
References ........................................................................................................ 30
2
, 4PAHPRM2 Research Methods 2 Data Concepts & Data Analysis
BSc Psychology Year 1
CHI-SQUARE TESTS
Chi-Square Tests
• The chi-square test is a non-parametric inferential statistical test for analysing categorical data
• The test examines the discrepancy between expected frequencies and observed frequencies
o We are interested in drawing conclusions about the distribution of scores for a single
categorical variable
Deciding Which Chi-Square Test to Use
Test Name When to Use Design Variables Involved
You have one categorical variable and One- 1 categorical (with ≥2
Chi-square
want to see if its distribution matches sample categories)
Goodness of Fit
expected values
Chi-square Test You want to test for an association Between- 2 categorical (with any
for Contingency between two categorical variables subjects number of levels)
Tables
You have paired categorical data Within- 1 dichotomous variable
McNemar’s Test
subjects measured twice
The Chi-Square (𝛘𝟐 ) Statistic
• A single value, expressing magnitude of the difference between the observed and expected values
• Usually, the bigger the difference between observed and expected values, the bigger χ"
($%&)!
χ" = Σ &
where O is the observed frequencies, and E is the expected frequencies
THE CHI-SQUARE GOODNESS-OF-FIT TEST
Degrees of Freedom
• The degrees of freedom for the chi-square goodness-of-fit test are always one less than the
number of categories
p-Value
• The probability of obtaining this value of χ" or larger, when sampling from a population that generates
our expected frequencies
• The p-value depends on the size of χ" and the degrees of freedom
o The larger χ" , the smaller the p-value
o The larger the df, the smaller the p-value
The Critical Value
• Whenever there is an χ" value above 3.84, the p-value is < .05
o Therefore, the observed values significantly differ from the expected values, so there is a
significant difference
3
