General Statistical Methods
There are several important concepts that we will adhere to in our group. These involve design considerations, execution considerations and analysis concerns. The standard for our field is null hypothesis significance testing, which means that we are generally comparing our data to a null hypothesis, generating an effect size and a p-value. As a general rule, we report both of these both within our Rmd/qmd scripts, and in our publications.
We generally use an <math display="inline">\alpha</math> of <math display="inline">p<0.05</math> to determine significance, which means that (if true) we are rejecting the null hypothesis. This is known as null hypothesis significance testing.
An alternative approach is to use a Bayesian approach, described in more detail in this document
Pairwise Testing
If you have two groups (and two groups only) that you want to know if they are different, you will normally want to do a pairwise test. This is not the case if you have paired data (before and after for example). The most common of these is something called a Student’s t-test, but this test has two key assumptions:
- The data are normally distributed
- The two groups have equal variance
Testing the Assumptions
Best practice is to first test for normality, and if that test passes, to then test for equal variance
Testing Normality
To test normality, we use a Shapiro-Wilk test (details on Wikipedia on each of your two groups). Below is an example where there are two groups:
<syntaxhighlight lang="r">#create seed for reproducibility
set.seed(1265)
test.data <- tibble(Treatment=c(rep("Experiment",6), rep("Control",6)),
Result = rnorm(n=12, mean=10, sd=3))
- test.data$Treatment <- as.factor(test.data$Treatment)
kable(test.data, caption="The test data used in the following examples")</syntaxhighlight>
The test data used in the following examples
| Treatment
|
Result
|
| Experiment
|
11.26
|
| Experiment
|
8.33
|
| Experiment
|
9.94
|
| Experiment
|
11.83
|
| Experiment
|
6.56
|
| Experiment
|
11.41
|
| Control
|
8.89
|
| Control
|
11.59
|
| Control
|
9.39
|
| Control
|
8.74
|
| Control
|
6.31
|
| Control
|
7.82
|
Each of the two groups, in this case Test and Control must have Shapiro-Wilk tests done separately. Some sample code for this is below (requires dplyr to be loaded):
<syntaxhighlight lang="r">#filter only for the control data
control.data <- filter(test.data, Treatment=="Control")
- The broom package makes the results of the test appear in a table, with the tidy command
library(broom)
- run the Shapiro-Wilk test on the values
shapiro.test(control.data$Result) %>% tidy %>% kable(caption="Shapiro-Wilk test for normality of control data")</syntaxhighlight>
Shapiro-Wilk test for normality of control data
| statistic
|
p.value
|
method
|
| 0.968
|
0.88
|
Shapiro-Wilk normality test
|
<syntaxhighlight lang="r">experiment.data <- filter(test.data, Treatment=="Experiment")
shapiro.test(test.data$Result) %>% tidy %>% kable(caption="Shapiro-Wilk test for normality of the test data")</syntaxhighlight>
Shapiro-Wilk test for normality of the test data
| statistic
|
p.value
|
method
|
| 0.93
|
0.377
|
Shapiro-Wilk normality test
|
Based on these results, since both p-values are >0.05 we do not reject the presumption of normality and can go on. If one or more of the p-values were less than 0.05 we would then use a Mann-Whitney test (also known as a Wilcoxon rank sum test) will be done, see below for more details.
Testing for Equal Variance
We generally use the car package which contains code for Levene’s Test to see if two groups can be assumed to have equal variance. For more details see @car:
<syntaxhighlight lang="r">#load the car package
library(car)</syntaxhighlight>
Loading required package: carData
The following object is masked from 'package:dplyr':
recode
<syntaxhighlight lang="r">#runs the test, grouping by the Treatment variable
leveneTest(Result ~ Treatment, data=test.data) %>% tidy %>% kable(caption="Levene's test on test data")</syntaxhighlight>
Warning in leveneTest.default(y = y, group = group, ...): group coerced to
factor.
Levene’s test on test data
| statistic
|
p.value
|
df
|
df.residual
|
| 0.368
|
0.558
|
1
|
10
|
Performing the Appropriate Pairwise Test
The logic to follow is:
- If the Shapiro-Wilk test passes, do Levene’s test. If it fails for either group, move on to a Wilcoxon Rank Sum Test.
- If Levene’s test passes, do a Student’s t Test, which assumes equal variance.
- If Levene’s test fails, do a Welch’s t Test, which does not assume equal variance.
Student’s t Test
<syntaxhighlight lang="r">#The default for t.test in R is Welch's, so you need to set the var.equal variable to be TRUE
t.test(Result~Treatment,data=test.data, var.equal=T) %>% tidy %>% kable(caption="Student's t test for test data")</syntaxhighlight>
Student’s t test for test data
| estimate
|
estimate1
|
estimate2
|
statistic
|
p.value
|
parameter
|
conf.low
|
conf.high
|
method
|
alternative
|
| -1.1
|
8.79
|
9.89
|
-0.992
|
0.345
|
10
|
-3.56
|
1.37
|
Two Sample t-test
|
two.sided
|
Welch’s t Test
<syntaxhighlight lang="r">#The default for t.test in R is Welch's, so you need to set the var.equal variable to be FALSE, or leave the default
t.test(Result~Treatment,data=test.data, var.equal=F) %>% tidy %>% kable(caption="Welch's t test for test data")</syntaxhighlight>
Welch’s t test for test data
| estimate
|
estimate1
|
estimate2
|
statistic
|
p.value
|
parameter
|
conf.low
|
conf.high
|
method
|
alternative
|
| -1.1
|
8.79
|
9.89
|
-0.992
|
0.345
|
9.72
|
-3.57
|
1.38
|
Welch Two Sample t-test
|
two.sided
|
Wilcoxon Rank Sum Test
<syntaxhighlight lang="r"># no need to specify anything about variance
wilcox.test(Result~Treatment,data=test.data) %>% tidy %>% kable(caption="Mann-Whitney test for test data")</syntaxhighlight>
Mann-Whitney test for test data
| statistic
|
p.value
|
method
|
alternative
|
| 12
|
0.394
|
Wilcoxon rank sum exact test
|
two.sided
|
Corrections for Multiple Observations
The best illustration I have seen for the need for multiple observation corrections is this cartoon from XKCD (see http://xkcd.com/882/):
Any conceptually coherent set of observations must therefore be corrected for multiple observations. In most cases, we will use the method of @Benjamini1995 since our p-values are not entirely independent. Some sample code for this is here:
<syntaxhighlight lang="r">p.values <- c(0.023, 0.043, 0.056, 0.421, 0.012)
data.frame(unadjusted = p.values, adjusted=p.adjust(p.values, method="BH")) %>%
kable(caption="Effects of adjusting p-values by the method of Benjamini-Hochberg")</syntaxhighlight>
Effects of adjusting p-values by the method of Benjamini-Hochberg
| unadjusted
|
adjusted
|
| 0.023
|
0.057
|
| 0.043
|
0.070
|
| 0.056
|
0.070
|
| 0.421
|
0.421
|
| 0.012
|
0.057
|
Session Information
<syntaxhighlight lang="r">sessionInfo()</syntaxhighlight>
R version 4.4.1 (2024-06-14)
Platform: x86_64-apple-darwin20
Running under: macOS Sonoma 14.7
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Detroit
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] car_3.1-2 carData_3.0-5 broom_1.0.6 dplyr_1.1.4 tidyr_1.3.1
[6] knitr_1.48
loaded via a namespace (and not attached):
[1] vctrs_0.6.5 cli_3.6.3 rlang_1.1.4 xfun_0.47
[5] purrr_1.0.2 generics_0.1.3 jsonlite_1.8.8 glue_1.7.0
[9] backports_1.5.0 htmltools_0.5.8.1 fansi_1.0.6 rmarkdown_2.28
[13] abind_1.4-8 evaluate_0.24.0 tibble_3.2.1 fastmap_1.2.0
[17] yaml_2.3.10 lifecycle_1.0.4 compiler_4.4.1 htmlwidgets_1.6.4
[21] pkgconfig_2.0.3 rstudioapi_0.16.0 digest_0.6.37 R6_2.5.1
[25] tidyselect_1.2.1 utf8_1.2.4 pillar_1.9.0 magrittr_2.0.3
[29] withr_3.0.1 tools_4.4.1
References
