Latest revision as of 00:12, 16 October 2024

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:

          Result = rnorm(n=12, mean=10, sd=3))

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):

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:

Loading required package: carData

Attaching package: 'car'

The following object is masked from 'package:dplyr':

    recode

Warning in leveneTest.default(y = y, group = group, ...): group coerced to
factor.

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

Welch’s t Test

Wilcoxon Rank Sum Test

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:

 kable(caption="Effects of adjusting p-values by the method of Benjamini-Hochberg")</syntaxhighlight>

Session Information

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