Applied Biostatistical Modeling
(with Jamovi)
Tom Jansen
August 18, 2020
1
,Contents
Contents 2
4 Descriptive statistics 4
4.1 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Drawing graphs 6
6 Pragmatic matters 7
6.1 Contingency tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 Introduction to probability 8
7.1 The binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7.2 Discrete Probability Distributions - The Poisson distribution . . . . . . . . . 8
7.3 Other types of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Estimating unknown quantities from a sample 10
8.1 Probability estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9 Hypothesis testing 11
9.1 P-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.2 The power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10 Categorical data analysis 13
10.1 Pearson’s test of χ2 goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . 13
10.2 The χ2 Test of Association (/independence) . . . . . . . . . . . . . . . . . . 14
11 Comparing two means 16
11.1 Student t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
11.2 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.3 Checking the normality of a sample . . . . . . . . . . . . . . . . . . . . . . . 18
11.4 Checking the equality of variances . . . . . . . . . . . . . . . . . . . . . . . . 18
11.5 Non-parametric t-tests: Wilcoxon Signed Ranks test and Mann-Whitney U
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2
,CONTENTS 3
12 Correlation and linear regression 22
12.1 Linear Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
12.2 Multiple linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.3 Quantifying the fit of the regression model . . . . . . . . . . . . . . . . . . . 23
12.4 The adjusted R2 value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.5 Tests for individual coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12.6 Calculating standardised regression coefficients . . . . . . . . . . . . . . . . . 24
12.7 Assumptions of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12.8 Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
12.9 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.10Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
12.11Multiple Regression Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 32
13 Comparing several means (one-way ANOVA) 34
13.1 Calculating ANOVA by hand . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13.2 Multiple comparisons - Post hoc tests . . . . . . . . . . . . . . . . . . . . . . 36
13.3 Assumptions of one-way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . 38
13.4 Checking the homogeneity of variance assumption . . . . . . . . . . . . . . . 38
13.5 Checking the normality assumption . . . . . . . . . . . . . . . . . . . . . . . 39
13.6 One-way Repeated measures ANOVA . . . . . . . . . . . . . . . . . . . . . . 40
13.7 Friedman non-parametric repeated measures ANOVA test . . . . . . . . . . 41
13.8 Randomized Complete Block Design (RCBD) . . . . . . . . . . . . . . . . . 42
13.9 Multi-Way Repeated Measures ANOVA . . . . . . . . . . . . . . . . . . . . . 43
13.10Mixed ANOVA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
13.11Relationship between ANOVA and Student-t . . . . . . . . . . . . . . . . . . 44
14 Factorial ANOVA 45
14.1 Factorial ANOVA 1: balanced designs, no interactions . . . . . . . . . . . . . 45
14.2 Factorial ANOVA 2: balanced designs, interactions allowed . . . . . . . . . . 48
14.3 ANOVA as a linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
14.4 Factorial ANOVA 3: unbalanced designs, with interaction . . . . . . . . . . . 51
,Chapter 4
Descriptive statistics
• N: the number of observations.
• Xi : the label of one observation.
• X̄: the mean.
• Mode: the mode of a sample is the value that occurs the most. Often calculated when
you have nominal data, because the mean and median are useless for those sorts of
variables.
Types of Data
There are two categories of data: Categorical (nominal and ordinal) and Continuous (scale
and numerics) data.
Measures of variability
• Range: difference between highest and lowest value. Can be robust or not (check
outliers). Full spread of the data, very vulnerable to outliers.
• Interquartile range: Difference between the 25th percentile and the 75th percentile of
the data. Middle half of data, pretty robust.
• Average Absolute Deviation: distance from mean: |Xi − X̄| divided by n.
• Variance: Squared variations are better than absolute variations, therefore we use the
variance: s2 . With formula V ar(X) = N1 N 2
i=1 (Xi − X̄) . Jamovi is using a slightly
P
different equation: N1−1 N 2
i=1 (Xi − X̄) .
P
• Standard Deviation: the square root q of the variance. This is also called the Root Mean
1 PN
Squared Deviation (RMSD): σ̂ = N −1 i=1 (Xi − X̄)2 . Rule of thumb: expect 68%
of the data to fall into 1 sd, 95% into 2 sd and 99.7% to fall into 3 sd. Is expressed in
the same units as the data. Most popular measure of variation.
4
,CHAPTER 4. DESCRIPTIVE STATISTICS 5
Standardized Scores
Describe a variable in terms of the overall distribution. You can calculated the standard
score, that is describe a variable in how much std’s it lays from the mean: zi = Xiσ̂−X̄ . Than
you can convert that to percentages with the rule of thumb or with tables. This is relative
to its own population. BUT it is possible to compare standardized scores across completely
different variables.
4.1 Exploratory Data Analysis
Data Remarks
Sample Size N Adequate? Small?
Location Mean, Median
Extremes Min, Max
Disperity SD, IQR
Symmetry Reasonable? Doubts? Check boxplot for differences in quartile
Normality Reasonable? Concerns? QQ-plot, Shapiro-Wilk’s p
Outliers Case #..
Others Small sample size? Therefore....
,Chapter 5
Drawing graphs
• Histogram: for interval or ratio scale.
• Boxplot: Used for getting the IQR.
• Stripchart: Is preffered over boxplot when N is small (<14).
• Violin plot: Similar to boxplot, but also show kernel probability density.
• Scatterplot: Direction, Form, Strength, Unusual features? Used to check for correla-
tion.
• QQ-plot: Checks for normality, data is normal if the QQ-plot approximates a straight
line.
You should always check for center, dispersion, symmetry and outliers.
6
, Chapter 6
Pragmatic matters
6.1 Contingency tables
The contingency table shows a table of raw frequencies. That is, a count of the total number
of cases for different combinations of levels of the specified variables. However, often you
want your data to be organised in terms of percentages as well as counts.
7
(with Jamovi)
Tom Jansen
August 18, 2020
1
,Contents
Contents 2
4 Descriptive statistics 4
4.1 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Drawing graphs 6
6 Pragmatic matters 7
6.1 Contingency tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 Introduction to probability 8
7.1 The binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7.2 Discrete Probability Distributions - The Poisson distribution . . . . . . . . . 8
7.3 Other types of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Estimating unknown quantities from a sample 10
8.1 Probability estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9 Hypothesis testing 11
9.1 P-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.2 The power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10 Categorical data analysis 13
10.1 Pearson’s test of χ2 goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . 13
10.2 The χ2 Test of Association (/independence) . . . . . . . . . . . . . . . . . . 14
11 Comparing two means 16
11.1 Student t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
11.2 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.3 Checking the normality of a sample . . . . . . . . . . . . . . . . . . . . . . . 18
11.4 Checking the equality of variances . . . . . . . . . . . . . . . . . . . . . . . . 18
11.5 Non-parametric t-tests: Wilcoxon Signed Ranks test and Mann-Whitney U
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2
,CONTENTS 3
12 Correlation and linear regression 22
12.1 Linear Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
12.2 Multiple linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.3 Quantifying the fit of the regression model . . . . . . . . . . . . . . . . . . . 23
12.4 The adjusted R2 value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.5 Tests for individual coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12.6 Calculating standardised regression coefficients . . . . . . . . . . . . . . . . . 24
12.7 Assumptions of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12.8 Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
12.9 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.10Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
12.11Multiple Regression Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 32
13 Comparing several means (one-way ANOVA) 34
13.1 Calculating ANOVA by hand . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13.2 Multiple comparisons - Post hoc tests . . . . . . . . . . . . . . . . . . . . . . 36
13.3 Assumptions of one-way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . 38
13.4 Checking the homogeneity of variance assumption . . . . . . . . . . . . . . . 38
13.5 Checking the normality assumption . . . . . . . . . . . . . . . . . . . . . . . 39
13.6 One-way Repeated measures ANOVA . . . . . . . . . . . . . . . . . . . . . . 40
13.7 Friedman non-parametric repeated measures ANOVA test . . . . . . . . . . 41
13.8 Randomized Complete Block Design (RCBD) . . . . . . . . . . . . . . . . . 42
13.9 Multi-Way Repeated Measures ANOVA . . . . . . . . . . . . . . . . . . . . . 43
13.10Mixed ANOVA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
13.11Relationship between ANOVA and Student-t . . . . . . . . . . . . . . . . . . 44
14 Factorial ANOVA 45
14.1 Factorial ANOVA 1: balanced designs, no interactions . . . . . . . . . . . . . 45
14.2 Factorial ANOVA 2: balanced designs, interactions allowed . . . . . . . . . . 48
14.3 ANOVA as a linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
14.4 Factorial ANOVA 3: unbalanced designs, with interaction . . . . . . . . . . . 51
,Chapter 4
Descriptive statistics
• N: the number of observations.
• Xi : the label of one observation.
• X̄: the mean.
• Mode: the mode of a sample is the value that occurs the most. Often calculated when
you have nominal data, because the mean and median are useless for those sorts of
variables.
Types of Data
There are two categories of data: Categorical (nominal and ordinal) and Continuous (scale
and numerics) data.
Measures of variability
• Range: difference between highest and lowest value. Can be robust or not (check
outliers). Full spread of the data, very vulnerable to outliers.
• Interquartile range: Difference between the 25th percentile and the 75th percentile of
the data. Middle half of data, pretty robust.
• Average Absolute Deviation: distance from mean: |Xi − X̄| divided by n.
• Variance: Squared variations are better than absolute variations, therefore we use the
variance: s2 . With formula V ar(X) = N1 N 2
i=1 (Xi − X̄) . Jamovi is using a slightly
P
different equation: N1−1 N 2
i=1 (Xi − X̄) .
P
• Standard Deviation: the square root q of the variance. This is also called the Root Mean
1 PN
Squared Deviation (RMSD): σ̂ = N −1 i=1 (Xi − X̄)2 . Rule of thumb: expect 68%
of the data to fall into 1 sd, 95% into 2 sd and 99.7% to fall into 3 sd. Is expressed in
the same units as the data. Most popular measure of variation.
4
,CHAPTER 4. DESCRIPTIVE STATISTICS 5
Standardized Scores
Describe a variable in terms of the overall distribution. You can calculated the standard
score, that is describe a variable in how much std’s it lays from the mean: zi = Xiσ̂−X̄ . Than
you can convert that to percentages with the rule of thumb or with tables. This is relative
to its own population. BUT it is possible to compare standardized scores across completely
different variables.
4.1 Exploratory Data Analysis
Data Remarks
Sample Size N Adequate? Small?
Location Mean, Median
Extremes Min, Max
Disperity SD, IQR
Symmetry Reasonable? Doubts? Check boxplot for differences in quartile
Normality Reasonable? Concerns? QQ-plot, Shapiro-Wilk’s p
Outliers Case #..
Others Small sample size? Therefore....
,Chapter 5
Drawing graphs
• Histogram: for interval or ratio scale.
• Boxplot: Used for getting the IQR.
• Stripchart: Is preffered over boxplot when N is small (<14).
• Violin plot: Similar to boxplot, but also show kernel probability density.
• Scatterplot: Direction, Form, Strength, Unusual features? Used to check for correla-
tion.
• QQ-plot: Checks for normality, data is normal if the QQ-plot approximates a straight
line.
You should always check for center, dispersion, symmetry and outliers.
6
, Chapter 6
Pragmatic matters
6.1 Contingency tables
The contingency table shows a table of raw frequencies. That is, a count of the total number
of cases for different combinations of levels of the specified variables. However, often you
want your data to be organised in terms of percentages as well as counts.
7
