One- and two-sample inference in rigr
Charles Wolock, Brian D. Williamson, and Scott S. Emerson
2022-12-18
Source:vignettes/one_and_two_sample_inference.Rmd
one_and_two_sample_inference.Rmd## rigr version 1.0.5: Regression, Inference, and General Data Analysis Tools in RThe rigr package replicates many of the basic
inferential functions from R’s stats package, with an eye
toward inference as taught in an introductory statistics class. To
demonstrate these basic functions, we will use the included
mri dataset. Information about the dataset can be found by
running ?mri. Since the data is part of the package, we can
load it via
data(mri)Throughout this vignette, we will assume familiarity with basic data manipulation and statistical tasks.
One and two-sample inference
Many of our analyses boil down to one-sample or two-sample problems, such as “What is the mean time to graduation?”, “What is the median home price in Seattle?”, or “What is the difference in mean time to a relapse event between the control and the treatment group?” There are many methods of analyzing one- and two-sample relationships, and in our package we have implemented three common approaches.
t-tests
We are often interested in making statements about the average (or mean) value of a variable. A one-sample t-test asks whether the mean of the distribution from which a sample is drawn is equal to some fixed value. A two-sample t-test asks whether the difference in means between two distributions is equal to some value (often zero, i.e., no difference in means).
Our function ttest() is flexible, allowing
stratification, calculation of the geometric mean, and equal/unequal
variances between samples. For example, a t-test of whether the mean of
the ldl variable is equal to 125 mg/dL can be performed
using rigr as follows:
ttest(mri$ldl, null.hypoth = 125)##
## Call:
## ttest(var1 = mri$ldl, null.hypoth = 125)
##
## One-sample t-test :
##
## Summary:
## Variable Obs Missing Mean Std. Err. Std. Dev. 95% CI
## mri$ldl 735 10 126 1.25 33.6 [123, 128]
##
## Ho: mean = 125 ;
## Ha: mean != 125
## t = 0.6433 , df = 724
## Pr(|T| > t) = 0.520256Note that in addition to running the hypothesis test,
ttest also returns a point estimate (the column
Mean under Summary) and a 95% confidence
interval for the true mean LDL.
If instead we wanted a two-sample t-test of whether the difference in
mean LDL between males and females were zero, we could stratify using
the by argument:
ttest(mri$ldl, by = mri$sex)##
## Call:
## ttest(var1 = mri$ldl, by = mri$sex)
##
## Two-sample t-test allowing for unequal variances :
##
## Summary:
## Group Obs Missing Mean Std. Err. Std. Dev. 95% CI
## mri$sex = Female 369 4 130.9 1.79 34.3 [127.4, 134.5]
## mri$sex = Male 366 6 120.6 1.69 32.1 [117.3, 123.9]
## Difference 735 10 10.3 2.47 <NA> [5.5, 15.2]
##
## Ho: difference in means = 0 ;
## Ha: difference in means != 0
## t = 4.194 , df = 721
## Pr(|T| > t) = 3.08428e-05In addition to using by, we can also run two-sample
tests by simply providing two data vectors:
ttest(mri$ldl[mri$sex == "Female"], mri$ldl[mri$sex == "Male"]).
Note that the default of ttest is to assume unequal
variances between groups, which we (the authors of this package) believe
to be the best choice in most scenarios. We also run two-sided tests by
default, but others can be specified, along with non-zero null
hypotheses, and tests at levels other than 0.95:
ttest(mri$ldl, null.hypoth = 125, conf.level = 0.9)##
## Call:
## ttest(var1 = mri$ldl, null.hypoth = 125, conf.level = 0.9)
##
## One-sample t-test :
##
## Summary:
## Variable Obs Missing Mean Std. Err. Std. Dev. 90% CI
## mri$ldl 735 10 126 1.25 33.6 [124, 128]
##
## Ho: mean = 125 ;
## Ha: mean != 125
## t = 0.6433 , df = 724
## Pr(|T| > t) = 0.520256
ttest(mri$ldl, by = mri$sex, var.eq = FALSE)##
## Call:
## ttest(var1 = mri$ldl, by = mri$sex, var.eq = FALSE)
##
## Two-sample t-test allowing for unequal variances :
##
## Summary:
## Group Obs Missing Mean Std. Err. Std. Dev. 95% CI
## mri$sex = Female 369 4 130.9 1.79 34.3 [127.4, 134.5]
## mri$sex = Male 366 6 120.6 1.69 32.1 [117.3, 123.9]
## Difference 735 10 10.3 2.47 <NA> [5.5, 15.2]
##
## Ho: difference in means = 0 ;
## Ha: difference in means != 0
## t = 4.194 , df = 721
## Pr(|T| > t) = 3.08428e-05If we prefer to run the test using summary statistics (sample mean,
sample standard deviation, and sample size) we can instead use the
ttesti function:
##
## Call:
## ttesti(obs = length(mri$weight), mean = mean(mri$weight), sd = sd(mri$weight),
## null.hypoth = 155)
##
## One-sample t-test :
##
## Summary:
## Obs Mean Std. Error Std. Dev. 95% CI
## var1 735 160 1.13 30.7 [158, 162]
##
## Ho: mean = 155 ;
## Ha: mean != 155
## t = 4.365 , df = 734
## Pr(|T| > t) = 1.45125e-05The result is the same as that provided by
ttest(mri$weight, null.hypoth = 155).
Tests of proportions
In the above example, we investigated the mean of a continuous random
variable. However, sometimes we work with binary data. In this case, we
may wish to make inference on probabilities. In rigr, we
can do this using proptest. For one-sample proportion
tests, there are both approximate (based on the normal distribution) and
exact (based on the binomial distribution) options. For example, we may
wish to test whether the proportion of LDL values that are greater than
128mg/dL is equal to 0.5.
proptest(mri$ldl > 128, null.hypoth = 0.5, exact = FALSE)##
## Call:
## proptest(var1 = mri$ldl > 128, exact = FALSE, null.hypoth = 0.5)
##
## One-sample proportion test (approximate) :
##
## Variable Obs Missing Estimate Std. Err. 95% CI
## mri$ldl > 128 735 10 0.4634483 0.0185 [0.427, 0.5]
## Summary:
##
## Ho: True proportion is = 0.5;
## Ha: True proportion is != 0.5
## Z = -1.97
## p-value = 0.049
proptest(mri$ldl > 128, null.hypoth = 0.5, exact = TRUE)##
## Call:
## proptest(var1 = mri$ldl > 128, exact = TRUE, null.hypoth = 0.5)
##
## One-sample proportion test (exact) :
##
## Variable Obs Missing Estimate Std. Err. 95% CI
## mri$ldl > 128 735 10 0.4634483 0.0185 [0.427, 0.501]
## Summary:
##
## Ho: True proportion is = 0.5;
## Ha: True proportion is != 0.5
##
## p-value = 0.0534Note that we are creating our binary data within the
proptest call. The proptest function works
with 0-1 numeric data, two-level factors, or (as above)
TRUE/FALSE data. Using the exact
argument allows us to choose what kind of test we run. In this case, the
results are quite similar.
Given two samples, we can also test whether two proportions are equal
to each other. There is no exact option for a two-sample
test. Here we test whether the proportion of men with LDL greater than
128 mg/dL is the same as the proportion of women.
proptest(mri$ldl > 128, by = mri$sex)##
## Call:
## proptest(var1 = mri$ldl > 128, by = mri$sex)
##
## Two-sample proportion test (approximate) :
##
## Group Obs Missing Mean Std. Err. 95% CI
## mri$sex = Female 369 4 0.5287671 0.0261 [0.4776, 0.58]
## mri$sex = Male 366 6 0.3972222 0.0258 [0.3467, 0.448]
## Difference 735 10 0.1315449 0.0367 [0.0596, 0.203]
## Summary:
##
## Ho: Difference in proportions = 0
## Ha: Difference in proportions != 0
## Z = 3.55
## p.value = 0.000383The proptesti function is analogous to
ttesti described above - rather than providing data
vectors, we can provide summary statistics in the form of counts of
successes out of a total number of trials. Here we test whether the
proportion of people with weight greater than 155 lbs is equal to
0.6.
##
## Call:
## proptesti(x1 = sum(mri$weight > 155), n1 = length(mri$weight),
## exact = FALSE, null.hypoth = 0.6)
##
## One-sample proportion test (approximate) :
##
## Variable Obs Mean Std. Error 95% CI
## var1 735 0.533 0.0184 [0.497, 0.569]
## Summary:
##
## Ho: True proportion is = 0.6;
## Ha: True proportion is != 0.6
## Z = -3.69
## p.value = 0.000225Wilcoxon and Mann-Whitney
The Wilcoxon and Mann-Whitney tests, which use the “rank” of the given variables, are nonparametric methods for analyzing the locations of the underlying distributions that gave rise to a dataset. They are often viewed as alternative to one- and two-sample t-tests, respectively.
Our function wilcoxon() takes one or two samples and
performs either an approximate or exact test of location. Since these
tests are not based on the mean of the data, the output looks slightly
different from that of ttest. Here, we perform a paired
(matched) test on made-up data comparing individuals with cystic
fibrosis (CF) to health individuals.
## create the data
cf <- c(1153, 1132, 1165, 1460, 1162, 1493, 1358, 1453, 1185, 1824, 1793, 1930, 2075)
healthy <- c(996, 1080, 1182, 1452, 1634, 1619, 1140, 1123, 1113, 1463, 1632, 1614, 1836)
wilcoxon(cf, healthy, paired = TRUE)##
## Wilcoxon signed rank test
## obs sum ranks expected
## positive 10 71 45.5
## negative 3 20 45.5
## zero 0 0 0.0
## all 13 91 91.0
##
## unadjusted variance 204.75
## adjustment for ties 0.00
## adjustment for zeroes 0.00
## adjusted variance 204.75
## H0 Ha
## Hypothesized Median 0 two.sided
## Test Statistic p-value
## Z 1.7821 0.074735This function can also provide a confidence interval for the median, although unlike the Wilcoxon and Mann-Whitney tests, this confidence interval is semiparametric rather than nonparametric.
wilcoxon(cf, healthy, paired = TRUE, conf.int = TRUE)##
## Wilcoxon signed rank test
## obs sum ranks expected
## positive 10 71 45.5
## negative 3 20 45.5
## zero 0 0 0.0
## all 13 91 91.0
##
## unadjusted variance 204.75
## adjustment for ties 0.00
## adjustment for zeroes 0.00
## adjusted variance 204.75
## H0 Ha
## Hypothesized Median 0 two.sided
## Test Statistic p-value CI Point Estimate
## Z 1.7821 0.074735 [-27, 238.5] 117.5Note that there is no version of wilcoxon using summary
statistics, since the test relies on the ranks of the observed data.