Table of Contents
Statistical techniques I (Z and Chi-squared tests)................................................................3
Dependent and independent measurements......................................................................3
One sample Z-test..............................................................................................................3
Assumptions:................................................................................................................................3
CI assumptions for one sample z-test...........................................................................................3
Chi-squared test- goodness of fit........................................................................................4
Two sample Z-test..............................................................................................................4
CI for two sample z-test................................................................................................................4
Relative risk and odds ratio................................................................................................5
Chi-squared for homogeneity (two-way tables)..................................................................5
Chi-squared for independence............................................................................................6
Assumptions.................................................................................................................................6
Statistical techniques II (T-tests).........................................................................................7
One sample t-test...............................................................................................................8
Dependent samples t-test..................................................................................................9
one-sample t-test of within-subject differences...........................................................................9
Independent sample t-test...............................................................................................10
Degrees of freedom....................................................................................................................10
CI for independent events..........................................................................................................10
2x2 design..................................................................................................................................10
Transformation of data....................................................................................................11
Outliers............................................................................................................................11
Statistical techniques III (correlation tests).......................................................................12
Correlation.......................................................................................................................12
The quantification of correlation coefficients.............................................................................12
Positive linear correlation...........................................................................................................12
Negative linear correlation.........................................................................................................12
No association............................................................................................................................12
Nonlinear correlations................................................................................................................12
Pearson’s product moment correlation: r.........................................................................13
Meaning of r...............................................................................................................................13
Interpretation of r......................................................................................................................13
Alternative to r...........................................................................................................................13
1
,T-test of association.........................................................................................................14
Assumptions...............................................................................................................................14
Limitations to correlations................................................................................................14
1. Measures of linear association...............................................................................................14
2. Sensitive to outliers................................................................................................................14
3. Extrapolation..........................................................................................................................15
4. Covariation does not mean causality......................................................................................15
Causality..........................................................................................................................15
Causality postulates.........................................................................................................15
Plausibility..................................................................................................................................15
Strength......................................................................................................................................15
Consistency................................................................................................................................15
Specificity...................................................................................................................................15
Temporality................................................................................................................................15
Biological gradient......................................................................................................................15
Coherence..................................................................................................................................16
Experiment (*** most important)..............................................................................................16
Analogy......................................................................................................................................16
Regression.......................................................................................................................16
Assumptions...............................................................................................................................16
From correlation to regression.........................................................................................16
Correlation.................................................................................................................................16
Regression..................................................................................................................................16
Non-parametric tests.......................................................................................................17
Spearman’s Rho rs............................................................................................................17
Wilcoxon sign ranked test................................................................................................17
Mann-Whitney U test.......................................................................................................18
Summary..........................................................................................................................18
Limitations.......................................................................................................................19
2
, Statistical techniques I (Z and Chi-squared tests)
Pr (data∨H 0 ) the probability of the data sample given that the H0 is true.
= very unlikely reject the H0
= likely retain the H0
point estimate−expected value
Test statistic=
standard error
Dependent and independent measurements
Dependent Independent
Within test subjects Comparison between 2 groups
Paired measurements Unpaired measurements
One sample Z-test
Dichotomous variable only
When the binomial distribution approaches a very large n with many low probabilities then
it is appropriate to replace the test statistic to z-test
X−π 0
Z=
SE
Where the SE in this case is σ calculated by σ =
√ π (1 – π )
n
**** with the Z-value it is important to realize that the table entry is to the left of the Z
value. Therefore, if it is Pr ( Z >0.63 )=1−0.7357 .
Assumptions:
n π 0 ≥ 10∧n(1−π ¿¿ 0)≥10 ¿
Then do a z-test instead. Where the π 0is the population proportion under H0
Population is larger than the sample (approximately x20)
Sample is random
CI assumptions for one sample z-test
The same as above
N success > 15 and non-successes > 15
Otherwise n+2/n+4 rule
If the p value lies within the CI intervals, then you retain the H0
If the p value lies outside of the CI intervals, then you reject the H0
3
Statistical techniques I (Z and Chi-squared tests)................................................................3
Dependent and independent measurements......................................................................3
One sample Z-test..............................................................................................................3
Assumptions:................................................................................................................................3
CI assumptions for one sample z-test...........................................................................................3
Chi-squared test- goodness of fit........................................................................................4
Two sample Z-test..............................................................................................................4
CI for two sample z-test................................................................................................................4
Relative risk and odds ratio................................................................................................5
Chi-squared for homogeneity (two-way tables)..................................................................5
Chi-squared for independence............................................................................................6
Assumptions.................................................................................................................................6
Statistical techniques II (T-tests).........................................................................................7
One sample t-test...............................................................................................................8
Dependent samples t-test..................................................................................................9
one-sample t-test of within-subject differences...........................................................................9
Independent sample t-test...............................................................................................10
Degrees of freedom....................................................................................................................10
CI for independent events..........................................................................................................10
2x2 design..................................................................................................................................10
Transformation of data....................................................................................................11
Outliers............................................................................................................................11
Statistical techniques III (correlation tests).......................................................................12
Correlation.......................................................................................................................12
The quantification of correlation coefficients.............................................................................12
Positive linear correlation...........................................................................................................12
Negative linear correlation.........................................................................................................12
No association............................................................................................................................12
Nonlinear correlations................................................................................................................12
Pearson’s product moment correlation: r.........................................................................13
Meaning of r...............................................................................................................................13
Interpretation of r......................................................................................................................13
Alternative to r...........................................................................................................................13
1
,T-test of association.........................................................................................................14
Assumptions...............................................................................................................................14
Limitations to correlations................................................................................................14
1. Measures of linear association...............................................................................................14
2. Sensitive to outliers................................................................................................................14
3. Extrapolation..........................................................................................................................15
4. Covariation does not mean causality......................................................................................15
Causality..........................................................................................................................15
Causality postulates.........................................................................................................15
Plausibility..................................................................................................................................15
Strength......................................................................................................................................15
Consistency................................................................................................................................15
Specificity...................................................................................................................................15
Temporality................................................................................................................................15
Biological gradient......................................................................................................................15
Coherence..................................................................................................................................16
Experiment (*** most important)..............................................................................................16
Analogy......................................................................................................................................16
Regression.......................................................................................................................16
Assumptions...............................................................................................................................16
From correlation to regression.........................................................................................16
Correlation.................................................................................................................................16
Regression..................................................................................................................................16
Non-parametric tests.......................................................................................................17
Spearman’s Rho rs............................................................................................................17
Wilcoxon sign ranked test................................................................................................17
Mann-Whitney U test.......................................................................................................18
Summary..........................................................................................................................18
Limitations.......................................................................................................................19
2
, Statistical techniques I (Z and Chi-squared tests)
Pr (data∨H 0 ) the probability of the data sample given that the H0 is true.
= very unlikely reject the H0
= likely retain the H0
point estimate−expected value
Test statistic=
standard error
Dependent and independent measurements
Dependent Independent
Within test subjects Comparison between 2 groups
Paired measurements Unpaired measurements
One sample Z-test
Dichotomous variable only
When the binomial distribution approaches a very large n with many low probabilities then
it is appropriate to replace the test statistic to z-test
X−π 0
Z=
SE
Where the SE in this case is σ calculated by σ =
√ π (1 – π )
n
**** with the Z-value it is important to realize that the table entry is to the left of the Z
value. Therefore, if it is Pr ( Z >0.63 )=1−0.7357 .
Assumptions:
n π 0 ≥ 10∧n(1−π ¿¿ 0)≥10 ¿
Then do a z-test instead. Where the π 0is the population proportion under H0
Population is larger than the sample (approximately x20)
Sample is random
CI assumptions for one sample z-test
The same as above
N success > 15 and non-successes > 15
Otherwise n+2/n+4 rule
If the p value lies within the CI intervals, then you retain the H0
If the p value lies outside of the CI intervals, then you reject the H0
3
