Paired Comparisons of Two Sleep Remedies, revisited
We revisit the sleep remedy dataset (from WS16) because the normality assumption of the sampling distribution of the mean did not seem to be reasonable. The plots below remind us of the data and bootstrap estimate of the sampling distribution.
Use the example in ADA1 Chapter 6 on Nonparametric Methods to state the hypothesis and results of the tests:
Sign test
(1 p) Hypotheses in words and notation. Be sure to specify which population parameter is being tested.
(1 p) R code and results for test (use the BSDA package and SIGN.test()). Use \(\alpha = 0.10\) (for a change).
(1 p) Conclusion based on test.
90% CI for the Median
(0.5 p) R code and CI (use the wilcox.test()).
(0.5 p) State the assumptions for the Wilcoxon CI.
(1 p) Determine the appropriate method for a CI (sign or Wilcoxon), then present and interpret the 90% CI with reference to the population parameter.
The Number of Breaks in Yarn during Weaving
This data set gives the number of warp breaks per loom, where a loom corresponds to a fixed length of yarn. We will analyze wool type B and see whether the tension affects the number of breaks.
A data frame with 54 observations on 3 variables.
[,1] breaks numeric The number of breaks
[,2] wool factor The type of wool (A or B)
[,3] tension factor The level of tension (L, M, H)
Notice that in the plot below I include the points, median, and IQR (the center 50%). Because there’s only 9 observations per group the boxplot and violin plot that I would typically also include becomes distracting and less informative. With very small sample sizes, the salience of individual values should increase relative to summaries.
# A set of useful summary functions is provided from the Hmisc package:#library(Hmisc)stat_sum_df <-function(fun, geom="crossbar", ...) {stat_summary(fun.data=fun, colour="gray60", geom=geom, width=0.2, ...)}# Plot the data using ggplotlibrary(ggplot2)p <-ggplot(dat_warp_sub, aes(x = tension, y = breaks))# plot a reference line for the global mean (assuming no groups)p <- p +geom_hline(yintercept =mean(dat_warp_sub$breaks),colour ="black", linetype ="dashed", size =0.3, alpha =0.5)# box for IQR and medianp <- p +stat_sum_df("median_hilow", fun.args =list(conf.int=0.5), show.legend=FALSE)# diamond at mean for each groupp <- p +stat_summary(fun = median, geom ="point", shape =18, size =4,colour ="red", alpha =0.8)# points for observed datap <- p +geom_point(alpha =0.5) #position = position_jitter(w = 0.05, h = 0), alpha = 1)p <- p +labs(title ="Number of breaks by loom tension\nfor wool B")p <- p +expand_limits(y =10)p <- p +theme_bw()#print(p)# log scalep2 <- pp2 <- p2 +scale_y_log10(breaks =seq(0, 50, by=10))p2 <- p2 +labs(y ="log10(breaks)")p2 <- p2 +expand_limits(y =10)#print(p2)library(gridExtra)
Attaching package: 'gridExtra'
The following object is masked from 'package:dplyr':
combine
grid.arrange(p, p2, ncol=2)
Some numerical summaries may be helpful later.
# summary of eachby(dat_warp_sub$breaks, dat_warp_sub$tension, summary)
dat_warp_sub$tension: L
Min. 1st Qu. Median Mean 3rd Qu. Max.
14.00 20.00 29.00 28.22 31.00 44.00
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dat_warp_sub$tension: M
Min. 1st Qu. Median Mean 3rd Qu. Max.
16.00 21.00 28.00 28.78 39.00 42.00
------------------------------------------------------------
dat_warp_sub$tension: H
Min. 1st Qu. Median Mean 3rd Qu. Max.
13.00 15.00 17.00 18.78 21.00 28.00
# A tibble: 3 × 6
tension mean sd median iqr n
<fct> <dbl> <dbl> <dbl> <dbl> <int>
1 L 28.2 9.86 29 11 9
2 M 28.8 9.43 28 18 9
3 H 18.8 4.89 17 6 9
Nonparametric methods are available to use even when assumptions for other tests are met. Use the Kruskal-Wallis rank sum test to assess whether there are differences in medians between these three groups. If there are, determine which pairs differ using the Wilcox-Mann-Whitney test.
Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot
compute exact p-value with ties
Wilcoxon rank sum test with continuity correction
data: breaks by tension
W = 41.5, p-value = 0.9647
alternative hypothesis: true location shift is not equal to 0
Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot
compute exact p-value with ties
Wilcoxon rank sum test with continuity correction
data: breaks by tension
W = 64.5, p-value = 0.03768
alternative hypothesis: true location shift is not equal to 0
Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot
compute exact p-value with ties
Wilcoxon rank sum test with continuity correction
data: breaks by tension
W = 67.5, p-value = 0.01897
alternative hypothesis: true location shift is not equal to 0
(1 p) Interpret the pairwise comparisons above (which pairs differ) at a Bonferroni-corrected significance level of \(\alpha/3 = 0.0333\).
(1 p) Summarize results by ordering the means and grouping pairs that do not differ (see notes for examples).
REPLACE THIS EXAMPLE WITH YOUR RESULTS.
Example: Groups A and C differ, but B is not different from either.
(These groups are ordered by mean, so A has the lowest median and C has the highest.)
Group A Group B Group C
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