Two proportions: pooled test, unpooled interval

Two samples, two proportions

With independent samples of sizes $n_1$ and $n_2$ giving ${\hat{p}}_1$ and ${\hat{p}}_2$, Module 11’s rule applies unchanged: the estimator ${\hat{p}}_1-{\hat{p}}_2$ has mean $p_1-p_2$ and a variance that adds,

$$\mathrm{Var}({\hat{p}}_1-{\hat{p}}_2)=\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}$$

What goes into that variance depends, once again, on whether a null value is available.

The test. $H_0:p_1=p_2$ says the two samples come from populations with the same proportion, call it $p$. The best estimate of that common value pools the counts:

$${\hat{p}}_c=\frac{x_1+x_2}{n_1+n_2},{\mathrm{SE}}_{\text{test}}=\sqrt{{\hat{p}}_c(1-{\hat{p}}_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)},z=\frac{{\hat{p}}_1-{\hat{p}}_2}{{\mathrm{SE}}_{\text{test}}}$$

The interval. An interval for $p_1-p_2$ assumes nothing about equality, so each $\hat{p}$ estimates its own variance:

$$({\hat{p}}_1-{\hat{p}}_2)\pm z_{\alpha /2}\sqrt{\frac{{\hat{p}}_1(1-{\hat{p}}_1)}{n_1}+\frac{{\hat{p}}_2(1-{\hat{p}}_2)}{n_2}}$$

Worked example — default rates in two regions

Grand River Credit reviews consumer loans in two regions. Region 1: 80 defaults in 400 loans, ${\hat{p}}_1=0.20$. Region 2: 45 in 300, ${\hat{p}}_2=0.15$. Do the default rates differ? $\alpha =0.05$.

Step 1. $H_0:p_1-p_2=0$, $H_a:p_1-p_2\ne 0$. Two-sided.

Step 2. ${\hat{p}}_c=125/700=0.1786$.

$${\mathrm{SE}}_{\text{test}}=\sqrt{0.1786\times 0.8214\times \left(\frac{1}{400}+\frac{1}{300}\right)}=\sqrt{0.14670\times 0.005833}=0.02925$$

$$z=\frac{0.20-0.15}{0.02925}=1.71$$

Step 3. Two-sided at 5 %: $\pm 1.96$. p-value: $2(1-\Phi (1.71))=2(1-0.9564)=0.0872$ (R, with the unrounded $z$: 0.0874).

Step 4. $1.71<1.96$; $0.087>0.05$: do not reject at 5 %. There is insufficient evidence that the default rates differ between the regions.

The 95 % interval. Unpooled SE:

$$\sqrt{\frac{0.20\times 0.80}{400}+\frac{0.15\times 0.85}{300}}=\sqrt{0.0004+0.000425}=0.02872$$

$$0.05\pm 1.96\times 0.02872=0.05\pm 0.0563=(-0.0063,0.1063)$$

With 95 % confidence, region 1’s default rate is between 0.6 points lower and 10.6 points higher than region 2’s. The interval straddles zero — consistent with the test, and more informative: the difference could plausibly be as large as ten points, which a lender would care about, and the samples are too small to pin it down.

Why two different SEs

The test’s SE is computed under $H_0$, and $H_0$ says the proportions are equal — so one pooled estimate of the shared $p$ is the right input, exactly as $p_0$ was in L12.3. The interval makes no such assumption. The two SEs are close here (0.0293 vs 0.0287) because ${\hat{p}}_1$ and ${\hat{p}}_2$ are close; they diverge when the proportions or the sample sizes differ a lot. Write the pooled one in the test and the unpooled one in the interval, and say why in one clause: pooled because $H_0$ asserts a common $p$.

Conditions for both: at least five successes and five failures in each sample — 80, 320, 45, 255. Fine.