Lecture 7 Is A different than B (t-tests etc)

Eamonn Mallon
20/09/2020

Occam's razor applied to statistical models

mod_diamond2 <- lm(lprice ~ lcarat + color + cut + clarity, data = diamonds2)
  • No point carrying out an analysis that is more complicated than it has to be
  • The tests we will look at in the next two sessions, the classical tests, deal with some of the most frequent types of analysis
  • e.g. men's height vs women's, height versus weight etc.
  • FYI the R code is an example of a linear model, more on those in BS1070/MB1080

Todays' tests

  • t test (comparing two sample means with normal residuals)
  • wilcoxon's test (comparing two sample means with non-normal residuals)

Student's t-test and Guinness

  • Guinness
  • Student was the pseudonym of W.S. Gosset (1876 - 1937)
  • Head Experimental Brewer, small-sample, stratified, and repeated balanced experiments on barley for proving the best yielding varieties
  • Gosset was a friend of both Pearson and Fisher, a noteworthy achievement, for each had a massive ego and a loathing for the other. He was a modest man who once cut short an admirer with this comment: “Fisher would have discovered it all anyway.”
  • Other awesome Guinness ads

The t-test

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  • how likely is it that the two sample means were drawn from populations with the same average?
  • calculate a test statistic
  • how likely that we obtain a test statistic this big or bigger if the null hypothesis is true
    • compare the calculated test statistic to the critical value which is calculated on the assumption that the null hypothesis is true
  • quick test: what is the null hypothesis when comparing two means?

The t-test

  • t = \( \frac{difference\, between\, two\, means}{standard\, error\, of\, the\, difference} \)
  • t = \( \frac{\bar{y}_A-\bar{y}_B}{S.E.D} \)
    • Lecture 3 explains the standard error of the mean (an estimate of how far the sample mean is likely to be from the population mean)
    • For two independent variables, the variance of a difference is the sum of the separate variances
    • \( S.E.M =\sqrt{\frac{s^2}{n}} \)
    • \( S.E.D =\sqrt{\frac{s_A^2}{n_A}+\frac{s_B^2}{n_B}} \)
  • t = \( \frac{\bar{y}_A-\bar{y}_B}{\sqrt{\frac{s_A^2}{n_A}+\frac{s_B^2}{n_B}}} \)

R code for a t-test

library(SMPracticals)#Data is in this package
t.test(formula = height ~ type,  # Formula
       data = darwin) # Dataframe containing the variables


    Welch Two Sample t-test

data:  height by type
t = 2.4371, df = 22.164, p-value = 0.02328
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.3909566 4.8423767
sample estimates:
mean in group Cross  mean in group Self 
           20.19167            17.57500

Outcrossed plants (mean +/- 95% confidence intervals: 20.19(0.39)) are larger than selfed plants (17.58 (4.48)) (t-test: t = 2.4371, df =22.164, p = 0.02328)

Wilcoxon test

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  • When the residuals are non-normal
  • Also know as a Mann-Whitney test
  • Rank all the data together
  • Add up the ranks for each treatment
  • compare the smaller value to a critical value

R code for a wilcoxon test

wilcox.test(formula = len ~ supp,  # Formula
       data = ToothGrowth, exact=FALSE) # Dataframe containing the variables


    Wilcoxon rank sum test with continuity correction

data:  len by supp
W = 575.5, p-value = 0.06449
alternative hypothesis: true location shift is not equal to 0

There is no significant difference between supplement types on their effect on tooth growth (Wilcoxon Rank-Sum Test: W= 575.5, n = 60, p = 0.06449)