lecture13

Prashant K

2024-02-27

Recap

Last class covered

  • How normal distributions are generated: Sum of random variables / mean of a random sample

    • Central limit theorem

  • Used R to simulate random events and explore distributions

Today’s class

  • Finish segment: How do I test if my data is normal? q-q plots and Shapiro-Wilk test

  • What questions do t-tests answer?

  • t-tests dissected: Layout of a general hypothesis test

  • Understand sampling distributions, standard error of mean (SEM) and confidence intervals (CI)

Checking for normality

How to check if my data is normally distributed?

  • We need metrics that give us the shape of the distribution without having to plot the distribution and staring at it

Quantifying the shape of the normal distribution

Mean +/- SD

Source: Wikipedia

Median, quartiles

quartiles divide the probability / distribution area into 4 quarters, a more general way is called quantile (note the spelling!)

Quantiles tell us about the distribution shape

Quartiles (4)

Deciles (10)

quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities. Source: Wikipedia

probability distribution = geom_density plot. Hence quantiles divide the range to make equal area under the geom_density curve

Q-Q plot enables comparison of distribution shapes

If they fall along the line, then it is normal.

Normal distribution

[1] 881 665

Non-normal distribution

[1] 1293    1

Formal Shapiro-Wilk test for normality

What is this testing for?

  • NULL hypothesis is that data is normally distributed
  • High p-value => Data is normal (hypothesis is accepted)

Normal distribution


    Shapiro-Wilk normality test

data:  .
W = 0.99807, p-value = 0.315

Non-normal distribution


    Shapiro-Wilk normality test

data:  life_exp
W = 0.95248, p-value < 2.2e-16

further reading: statology.org

What are t-tests for?

To make statistical conclusions of this kind based on data

  • “Smoking causes cancer”

But causality is really really hard to establish, that is why we talk about correlation/association

  • “Smoking is associated with cancer”

  • “Smokers are statistically significantly at a higher risk of having cancer”

“Evidence linking smoking and cancer appeared in the 1920s.” Source: The cigarette controversy, 2007/section: Smoking Causes Cancer: When Did They Know?

What data can test the smoking hypothesis?

2d data

  • Correlation

  • Regression (linear/non-linear)

1D data

  • t-test

Comparing two numbers vs two samples

Looking at the mean only is not enough:

3.15 is clearly > 2.46, does that alone satisfy the hypothesis?

  • We need to account for both the mean and variability, ie) the spread around the mean

This is what the t-statistic encompasses. The t-test tests for a hypothesis based on this statistic

t-tests finds statistical support against a NULL hypothesis

Null hypothesis in english - “smoking is NOT associated with cancer”

In statistical terms - “the samples of smokers and non-smokers came from the same population”

  • (assumption) For a t-test, this source population is normally distributed

  • (Other assumption for Student’s t-test) Equal variance between smoker dataset and non-smoker dataset (this can be violated for the welch t-test)

t-test is calculating difference in means / spread around mean

Watch full video here: “Student’s t-test” : Bozeman science/youtube

Sampling distributions

  1. Sample multiple times from a population (too expensive for a real experiment, so we imagine this)
  2. Calculate one statistic from the sample (such as mean) and record it (for each sample)
  3. Plot the statistic across all samples. The statistic is a random variable too dependent on the random sample, hence results in a different value for each sample.
  4. This results in a distribution (called sampling distribution of mean/statistic)
    • mean is nice because it follows a normal distribution (due to the central limit theorem!)
      • it’s mean ~ close approximation to the population mean
      • it’s standard deviation = standard error of mean (S.E.M.) ~ \(\sigma\)
  5. The properties of this distribution are helpful to understand the population which is what we are interested in

Sampling from a population of coloured balls

We will walk through this activity to understand sampling well

Let us see the sampling distribution

Calculating the sample mean, SEM, CI

Relating this to a 1-sample t-test

References

Great explanation of sample size in 7.2 (FIGURE 7.12) in moderndive/chap7

Summary

  • q-q plots and Shapiro-Wilk test enable checking data for normality

  • t-tests answer questions of association such as ‘smoking causes cancer’

  • t-test = p-value gives the probability that the NULL hypothesis: that both data groups were sampled from the same distribution is TRUE

  • Understood sampling distributions, standard error of mean (SEM) and confidence intervals (CI)