2021/2022 Statistical
Methods in
Research
An extensive summary of
Research Training II
G. Rozenblad
ERASMUS UNIVERSITY ROTTERDAM – PREMASTER
INTERNATIONAL BUSINESS ADMINISTRATION
,Table of Contents
Module 1: Bivariate Analysis: contingency tables ............................................................ 4
Analysis of associations between two qualitative variables ........................................ 4
Background Pearson's χ²-statistic for given distributions ............................................... 6
Application 1: given univariate distribution ................................................................ 6
Application 2: given bivariate distribution ................................................................ 11
Application Pearson's χ²-statistic for analysing independence ................................. 15
Problem of analysing independence ....................................................................... 15
Application 3: analysis of independence ................................................................ 17
Peculiarities contingency tables ................................................................................... 21
Analysis contingency tables in R ............................................................................... 21
Minimum required sample size................................................................................... 21
Yates or continuity correction .................................................................................... 24
Fisher's exact test ......................................................................................................... 25
Testing with p-values ................................................................................................... 26
Nature and extent of associations, reporting .......................................................... 27
Module 2: Bivariate Analysis: scatterplot, covariance and correlation ....................... 32
Scatterplots ...................................................................................................................... 33
Sample covariance and Pearson's correlation coefficient ....................................... 36
Sample covariance .................................................................................................... 36
Pearson's sample correlation coefficient ................................................................. 37
Background Pearson's correlation coefficient and testing ........................................ 39
Theoretical background correlation coefficient ..................................................... 39
Testing statistical independence ............................................................................... 42
Peculiarities Pearson's correlation coefficient ............................................................. 45
Pearson's correlation coefficient in R ........................................................................ 45
Nature and extent of association, degree of explanation, reporting................... 48
Testing with p-values ................................................................................................... 49
Testing for general hypotheses .................................................................................. 50
Non-linear dependencies .......................................................................................... 53
Impact of outliers ......................................................................................................... 54
Spearman's rank correlation coefficient .................................................................. 55
Spurious correlation and the partial correlation coefficient ...................................... 56
Background ................................................................................................................. 56
Where and when does spurious correlation occur ................................................. 58
1
, Analysis of spurious correlation .................................................................................. 60
Module 3: Bivariate Analysis: t-test ................................................................................... 64
Associations between quantitative variables and dichotomous qualitative
variables: t-test ................................................................................................................ 64
t-test: theoretical background ...................................................................................... 67
Analysis of statistical independence ......................................................................... 67
Assumptions ................................................................................................................. 69
Difference between sample averages .................................................................... 71
Ratio of sample variances .......................................................................................... 74
t-test: application ............................................................................................................ 75
Testing equality of variances...................................................................................... 75
Testing equality of means ........................................................................................... 78
Estimation of differences between means ............................................................... 80
t-test: peculiarities ........................................................................................................... 81
Application of t-tests in R ............................................................................................ 81
Nature and extent of association, reporting ............................................................ 83
General form of the t-test ........................................................................................... 84
Departures from normality: distribution-free tests .................................................... 86
Independent and dependent samples.................................................................... 87
Module 4: Bivariate Analysis: analysis of variance ......................................................... 91
Analysis of variance: main idea .................................................................................... 91
One-way Anova: theoretical background .................................................................. 96
Assumptions ................................................................................................................. 97
Calculation example ................................................................................................ 103
Properties of the Anova F-statistic ........................................................................... 104
Anova table ............................................................................................................... 107
One-way Anova: application ..................................................................................... 107
Hypothesis testing ...................................................................................................... 107
Estimation of pairwise differences between means.............................................. 108
Two-way Anova ............................................................................................................ 115
Background ............................................................................................................... 115
Assumptions ............................................................................................................... 118
Application hypothesis testing ................................................................................. 125
Analysis of variance: peculiarities ............................................................................... 127
Analysis of variance in R ........................................................................................... 127
Nature and extent of associations, reporting ........................................................ 130
2
, Assumed equality of variances................................................................................ 133
Distribution-free hypothesis tests .............................................................................. 135
Testing specific hypotheses ...................................................................................... 136
Dependent and independent samples ................................................................. 138
3
Methods in
Research
An extensive summary of
Research Training II
G. Rozenblad
ERASMUS UNIVERSITY ROTTERDAM – PREMASTER
INTERNATIONAL BUSINESS ADMINISTRATION
,Table of Contents
Module 1: Bivariate Analysis: contingency tables ............................................................ 4
Analysis of associations between two qualitative variables ........................................ 4
Background Pearson's χ²-statistic for given distributions ............................................... 6
Application 1: given univariate distribution ................................................................ 6
Application 2: given bivariate distribution ................................................................ 11
Application Pearson's χ²-statistic for analysing independence ................................. 15
Problem of analysing independence ....................................................................... 15
Application 3: analysis of independence ................................................................ 17
Peculiarities contingency tables ................................................................................... 21
Analysis contingency tables in R ............................................................................... 21
Minimum required sample size................................................................................... 21
Yates or continuity correction .................................................................................... 24
Fisher's exact test ......................................................................................................... 25
Testing with p-values ................................................................................................... 26
Nature and extent of associations, reporting .......................................................... 27
Module 2: Bivariate Analysis: scatterplot, covariance and correlation ....................... 32
Scatterplots ...................................................................................................................... 33
Sample covariance and Pearson's correlation coefficient ....................................... 36
Sample covariance .................................................................................................... 36
Pearson's sample correlation coefficient ................................................................. 37
Background Pearson's correlation coefficient and testing ........................................ 39
Theoretical background correlation coefficient ..................................................... 39
Testing statistical independence ............................................................................... 42
Peculiarities Pearson's correlation coefficient ............................................................. 45
Pearson's correlation coefficient in R ........................................................................ 45
Nature and extent of association, degree of explanation, reporting................... 48
Testing with p-values ................................................................................................... 49
Testing for general hypotheses .................................................................................. 50
Non-linear dependencies .......................................................................................... 53
Impact of outliers ......................................................................................................... 54
Spearman's rank correlation coefficient .................................................................. 55
Spurious correlation and the partial correlation coefficient ...................................... 56
Background ................................................................................................................. 56
Where and when does spurious correlation occur ................................................. 58
1
, Analysis of spurious correlation .................................................................................. 60
Module 3: Bivariate Analysis: t-test ................................................................................... 64
Associations between quantitative variables and dichotomous qualitative
variables: t-test ................................................................................................................ 64
t-test: theoretical background ...................................................................................... 67
Analysis of statistical independence ......................................................................... 67
Assumptions ................................................................................................................. 69
Difference between sample averages .................................................................... 71
Ratio of sample variances .......................................................................................... 74
t-test: application ............................................................................................................ 75
Testing equality of variances...................................................................................... 75
Testing equality of means ........................................................................................... 78
Estimation of differences between means ............................................................... 80
t-test: peculiarities ........................................................................................................... 81
Application of t-tests in R ............................................................................................ 81
Nature and extent of association, reporting ............................................................ 83
General form of the t-test ........................................................................................... 84
Departures from normality: distribution-free tests .................................................... 86
Independent and dependent samples.................................................................... 87
Module 4: Bivariate Analysis: analysis of variance ......................................................... 91
Analysis of variance: main idea .................................................................................... 91
One-way Anova: theoretical background .................................................................. 96
Assumptions ................................................................................................................. 97
Calculation example ................................................................................................ 103
Properties of the Anova F-statistic ........................................................................... 104
Anova table ............................................................................................................... 107
One-way Anova: application ..................................................................................... 107
Hypothesis testing ...................................................................................................... 107
Estimation of pairwise differences between means.............................................. 108
Two-way Anova ............................................................................................................ 115
Background ............................................................................................................... 115
Assumptions ............................................................................................................... 118
Application hypothesis testing ................................................................................. 125
Analysis of variance: peculiarities ............................................................................... 127
Analysis of variance in R ........................................................................................... 127
Nature and extent of associations, reporting ........................................................ 130
2
, Assumed equality of variances................................................................................ 133
Distribution-free hypothesis tests .............................................................................. 135
Testing specific hypotheses ...................................................................................... 136
Dependent and independent samples ................................................................. 138
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