General Statistics
Contents
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 scripts, and in our publications.
We generally use an <math>\alpha</math> of <math>p<0.05</math> to determine significance, which means that (if true) we are rejecting the null hypothesis.
Experimental Design
Where possible, prior to performing an experiment or study perform a power analysis. This is mainly to determine the appropriate sample sizes. To do this, you need to know a few of things:
- Either the sample size or the difference. The difference is provided in standard deviations. This means that you need to know the standard deviation of your measurement in question. It is a good idea to keep a log of these for your data, so that you can approximate what this is. If you hope to detect a correlation you will need to know the expected correlation coefficient.
- The desired false positive rate (normally 0.05). This is the rate at which you find a difference where there is none. This is also known as the type I error rate.
- The desired power (normally 0.8). This indicates that 80% of the time you will detect the effect if there is one. This is also known as 1 minus the false negative rate or 1 minus the Type II error rate.
We use the R package pwr to do a power analysis (Champely, 2020). Here is an example:
Pairwise Comparasons
require(pwr) false.negative.rate <- 0.05 statistical.power <- 0.8 sd <- 3.5 #this is calculated from known measurements difference <- 3 #you hope to detect a difference pwr.t.test(d = difference, sig.level = false.negative.rate, power=statistical.power)
## ## Two-sample t test power calculation ## ## n = 3.07 ## d = 3 ## sig.level = 0.05 ## power = 0.8 ## alternative = two.sided ## ## NOTE: n is number in *each* group
This tells us that in order to see a difference of at least 3, with at standard devation of 3.5 we need at least 3 observations in each group.
Correlations
The following is an example for detecting a correlation.
require(pwr) false.negative.rate <- 0.05 statistical.power <- 0.8 correlation.coefficient <- 0.6 #note that this is the r, to get the R2 value you will have to square this result. pwr.r.test(r = correlation.coefficient, sig.level = false.negative.rate, power=statistical.power)
## ## approximate correlation power calculation (arctangh transformation) ## ## n = 18.6 ## r = 0.6 ## sig.level = 0.05 ## power = 0.8 ## alternative = two.sided
This tells us that in order to detect a correlation coefficient of at least 0.6 (or an R^2 of 0.36) you need more than 18 observations.
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:
#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")
| 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):
#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$Result) %>% tidy %>% kable
| statistic | p.value | method |
|---|---|---|
| 0.968 | 0.88 | Shapiro-Wilk normality test |
experiment.data <- filter(test.data, Treatment=="Experiment") shapiro.test(test.data$Result) %>% tidy %>% kable
| 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:
#load the car package library(car) #runs the test, grouping by the Treatment variable leveneTest(Result ~ Treatment, data=test.data) %>% tidy %>% kable
| 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
#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
| 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
#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
| 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
# no need to specify anything about variance wilcox.test(Result~Treatment,data=test.data) %>% tidy %>% kable
| 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 Benjamini and Hochberg since our p-values are not entirely independent. Some sample code for this is here:
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"))
## unadjusted adjusted ## 1 0.023 0.0575 ## 2 0.043 0.0700 ## 3 0.056 0.0700 ## 4 0.421 0.4210 ## 5 0.012 0.0575
Session Information
sessionInfo()
## R version 4.4.1 (2024-06-14) ## Platform: x86_64-apple-darwin20 ## Running under: macOS Monterey 12.7.6 ## ## 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 ## [4] pwr_1.3-0 knitcitations_1.0.12 dplyr_1.1.4 ## [7] tidyr_1.3.1 knitr_1.48 ## ## loaded via a namespace (and not attached): ## [1] jsonlite_1.8.8 compiler_4.4.1 tidyselect_1.2.1 Rcpp_1.0.13 ## [5] xml2_1.3.6 stringr_1.5.1 jquerylib_0.1.4 yaml_2.3.10 ## [9] fastmap_1.2.0 R6_2.5.1 plyr_1.8.9 generics_0.1.3 ## [13] backports_1.5.0 tibble_3.2.1 RefManageR_1.4.0 lubridate_1.9.3 ## [17] bslib_0.8.0 pillar_1.9.0 rlang_1.1.4 utf8_1.2.4 ## [21] stringi_1.8.4 cachem_1.1.0 xfun_0.46 sass_0.4.9 ## [25] bibtex_0.5.1 timechange_0.3.0 cli_3.6.3 withr_3.0.0 ## [29] magrittr_2.0.3 digest_0.6.36 lifecycle_1.0.4 vctrs_0.6.5 ## [33] evaluate_0.24.0 glue_1.7.0 abind_1.4-5 fansi_1.0.6 ## [37] rmarkdown_2.27 purrr_1.0.2 httr_1.4.7 tools_4.4.1 ## [41] pkgconfig_2.0.3 htmltools_0.5.8.1
References
<a name=bib-pwr></a>[1] S. Champely. pwr: Basic Functions for Power Analysis. R package version 1.3-0. 2020. URL: https://CRAN.R-project.org/package=pwr.
