# Other two sample comparisons ST551 Lecture 25

## So far

Our two sample comparisons have focused on means (or proportions)

What else could we compare?

• Medians
• Variances
• Whole distributions

# Comparing medians: Mood’s median test

## Mood’s median test

Setting: two indpendent samples

$$Y_i$$ i.i.d sample of size $$n$$ from popuation with c.d.f $$F_Y$$
$$X_i$$ i.i.d sample of size $$m$$ from popuation with c.d.f $$F_X$$

$$m_Y = F_Y^{-1}(0.5)=$$ median of population that $$Y$$ is sampled from.
$$m_X = F_X^{-1}(0.5)=$$ median of population that $$X$$ is sampled from.

Comparison of interest: Is $$m_Y$$ the same as $$m_X$$?

## Example

A study is performed to assess the effect of fish oil supplements on diastolic blood pressure

• 25 subjects are randomly assigned to receive fish oil ($$n_Y = 12$$) or regular vegetable oil ($$n_X = 13$$) for two weeks.
• Each subject’s decrease in diastolic blood pressure over those two weeks is recorded (bigger numbers => better reduction in blood pressure)

Fish oil: -2.2, -0.8, 3.7, 4.9, 5, 5.2, 5.3, 6, 8, 8, 10.4 and 14

Regular oil: -6.4, -6.4, -5.9, -5.8, -5.3, -4.9, -4.4, 0.2, 2.1, 2.5, 2.5, 6.1 and 8.9

Question: Is the median blood pressure reduction the same for these two treatments?

• If the null is true, $$m_Y = m_X = m$$, what is our best guess for the median $$m$$?

• If the null is true, what proportion of the sample from $$Y$$ should be larger than $$m$$?

• If the null is true, what proportion of the sample from $$X$$ should be larger than $$m$$?

## Estimating the combined median

$\hat{m}_Y = \hat{m}_X = \hat{m} = \text{median}(Y_1, Y_2, \ldots, Y_n, X_1, X_2, \ldots, X_m)$

If the null is true, this estimate is an unbiased and consistent estimate of the common median, $$m$$.

We expect $$P(Y_i > m) = P(X_i > m)$$.

## Mood’s median test

Procedure:

1. Find the combined median $$\hat{m}$$.
2. Test the true proportion of Y’s greater than $$\hat{m}$$ is equal to the true proprtion of X’s greater than $$\hat{m}$$.

• Z-test for proportions/Chi-square test or Fishers exact test

## Example cont.

Combined sample:

##  [1] -6.4 -6.4 -5.9 -5.8 -5.3 -4.9 -4.4 -2.2 -0.8
## [10]  0.2  2.1  2.5  2.5  3.7  4.9  5.0  5.2  5.3
## [19]  6.0  6.1  8.0  8.0  8.9 10.4 14.0

Combined median, $$\hat{m}$$ = 2.5

Number $$> \hat{m}$$ Number $$\le \hat{m}$$
Fish Oil 10 2
Regular Oil 2 11

## Example cont.

\begin{aligned} Z &= \frac{\hat{p}_Y - \hat{p}_X}{\sqrt{\hat{p}_c(1 - \hat{p}_c) \left(\frac{1}{n} + \frac{1}{m}\right)}} \\ &= \frac{\frac{10}{12} - \frac{2}{13}}{\sqrt{\frac{12}{25}(1 - \frac{12}{25}) \left(\frac{1}{12} + \frac{1}{13}\right)}} \\ &= 3.4 \end{aligned}

p-value = $$6.8\times 10^{-4}$$.

There is convincing evidence that the median BP reduction on fish oil is different to the median BP reduction on regular oil.

# Wilcoxon Rank Sum test

## Wilcoxon Rank Sum

Wilcoxon Rank Sum, a.k.a Mann-Whitney U-test

Often presented as a test for equality of medians, like Wilcoxon Signed Rank, this isn’t true without further assumptions.

## Wilcoxon Rank Sum Procedure

1. Combine the samples
2. Rank the observations in the combined sample from smallest (1) to largest ($$n+m$$). If there are ties, assign the average rank to the tied observations.
3. Test statistic: Sum of the ranks in the sample with the smaller sample size
4. p-value: either use Normal approximation, or via permutation

Intutition: if all the observations come from the same distribution, it would be unlikely for all the observations in the samller sample to have all the highest ranks (or lowest).

## Example

Combined sample:

## Regular Oil Regular Oil Regular Oil Regular Oil
##        -6.4        -6.4        -5.9        -5.8
## Regular Oil Regular Oil Regular Oil    Fish Oil
##        -5.3        -4.9        -4.4        -2.2
##    Fish Oil Regular Oil Regular Oil Regular Oil
##        -0.8         0.2         2.1         2.5
## Regular Oil    Fish Oil    Fish Oil    Fish Oil
##         2.5         3.7         4.9         5.0
##    Fish Oil    Fish Oil    Fish Oil Regular Oil
##         5.2         5.3         6.0         6.1
##    Fish Oil    Fish Oil Regular Oil    Fish Oil
##         8.0         8.0         8.9        10.4
##    Fish Oil
##        14.0
## [1] 208

## Problems

• Location-shift assumption
• Not location shift