Assumptions

Assumptions

• The \(t\) procedures we have seen so far come with assumption of normally-distributed data
• but how much does that normality matter?
• Central Limit Theorem says that sampling distribution of sample mean is “approximately normal” if sample size is “large”.
• Hence same applies to difference of two sample means.
• How to use this in practice? Draw a picture and make a call about whether sample size large enough.

Blue Jays attendances

ggplot(jays, aes(sample = attendance)) + 
  stat_qq() + stat_qq_line()

Comments

• Distribution of attendances somewhat skewed to the right (because of the short lower tail and the sort-of curve)
• Sample size \(n = 25\) is reasonably large in Central Limit Theorem terms
• Use of \(t\) may be OK here despite skewed shape.

Learning to read

• Make normal quantile plots, one for each sample:

ggplot(kids, aes(sample = score)) + 
  stat_qq() + stat_qq_line() +
  facet_wrap(~ group)

Comments

• with sample sizes over 20 in each group, these are easily normal enough to use a \(t\)-test.
• the (sampling distribution of the) difference between two sample means tends to have a more normal distribution than either sample mean individually, so that two-sample \(t\) tends to be better than you’d guess.

Pain relief

• With matched pairs, assumption is of normality of differences, so work those out first:

pain %>% mutate(diff = druga - drugb) -> pain
pain

Normality of differences

ggplot(pain,aes(sample=diff)) + stat_qq() + stat_qq_line()

Comments

• This is very non-normal (the low outlier)
• The sample size of \(n = 12\) is not large
• We should have concerns about our matched pairs \(t\)-test.

Doing things properly

• The right way to use a \(t\) procedure:
  • draw a graph of our data (one of the standard graphs, or normal quantile plot)
  • use the graph to assess sufficient normality given the sample size
  • for a two-sample test, assess equality of spreads (boxplot easier for this)
  • if necessary, express our doubts about the \(t\) procedure (for now), or do a better test (later).

Looking ahead

• Looking at a normal quantile plot and assessing it with the sample size seems rather arbitrary. Can we do better? (Yes: using the bootstrap, later.)
• What to do if the \(t\) procedure is not to be trusted? Use a different test (later):
  • one sample: sign test
  • two samples: Mood’s median test
  • matched pairs: sign test on differences.
• If you have heard about the signed rank or rank sum tests: they come with extra assumptions that are usually not satisfied if normality fails.