Unstratified and Stratified Miettinen and Nurminen Test

Confidence interval

The confidence interval is based on the M&N method and given by the roots for PD = P₁ − P₀ of the equation:

$$\chi_\alpha^2 = \frac{(\hat{p}_1-\hat{p}_0-PD)^2}{\tilde{V}}$$,

where p̂₁ and p̂₀ are the observed values of P₁ and P₀, respectively;

• χ_α² = the upper cut point of size α from the central chi-square distribution with 1 degree of freedom (χ_α² = 3.84 for 95% confidence interval);

• PD = the difference between two population proportions (PD = P₁ − P₀);

$$\tilde{V}=\bigg[\frac{\tilde{p}_1(1-\tilde{p}_1)}{n_1}+ \frac{\tilde{p}_0(1-\tilde{p}_0)}{n_0}\bigg]\frac{n_1+n_0}{n_1+n_0-1}$$;

• n₁ and n₀ are the sample sizes for the test and control group, respectively;

• p̃₁ = maximum likelihood estimate of proportion on test computed as p̃₀ + PD;

• p̃₀ = maximum likelihood estimate of proportion on control under the constraint p̃₁ − p̃₀ = PD.

As stated above the 2-sided 100(1 − α)% CI is given by the roots for PD = P₁ − P₀. The bisection algorithm is used in the function to obtain the two roots (confidence interval) for PD.

p-value and Z-statistic

The Z-statistic is computed as:

$$Z_\text{diff}=\frac{\hat{p}_1-\hat{p}_0+S_0}{\sqrt{\tilde{V}}}$$ where p̂₁ and p̂₀ are the observed values for P₁ and P₀ respectively, S₀ is pre-specified proportion difference under the null;

• p̃₁ = maximum likelihood estimate of proportion on test computed as p̃₀ + S₀;

• p̃₀ = maximum likelihood estimate of proportion on control under the constraint p̃₁ − p̃₀ = S₀.

• For non-inferiority or one-sided equivalence hypothesis with S₀ > 0, the p-value, Pr (Z ≥ Z_diff | H₀), is computed based on Z_diff using the standard normal distribution.

• For non-inferiority or one-sided equivalence hypothesis with S₀ < 0, the p-value, Pr (Z ≤ Z_diff | H₀), is computed based on Z_diff using the standard normal distribution.

• For two-sided superiority test, the p-value Pr (χ_diff² ≤ χ₁² | H₀), is computed based on χ_diff² using the chi-square distribution with 1 degree of freedom, where χ_diff² = Z_diff².

Confidence interval

The confidence interval is based on the M&N method and given by the roots for PD = P₁ − P₀ of the equation:

$$\chi_\alpha^2 = \frac{(\hat{p}_1^*-\hat{p}_0^*-PD)^2}{\sum_{i=1}^I(W_i/\sum_{k=1}^{K}W_k)^2\tilde{V}_i}$$,

where $\hat{p}_s^* = \sum_{i=1}^I(W_i/\sum_{k=1}^KW_k)\hat{p}_{s i}$ for s = 0, 1;

$$\tilde{V}_i=\bigg[\frac{\tilde{p}_{1i}(1-\tilde{p}_{1i})}{n_{1i}}+\frac{\tilde{p}_{0i}(1-\tilde{p}_{0i})}{n_{0i}}\bigg]\frac{n_{1i}+n_{0i}}{n_{1i}+n_{0i}-1}$$;

• W_i is the weight for the i-th strata;
• $I = K = $ number of strata, i = k= strata;
• n_1i and n_0i are the sample sizes in i-th strata for the test and control group, respectively;
• p̂_1i and p̂_0i = observed proportion in i-th strata for the test and control, respectively;
• p̃_0i and p̃_1i are MLE for P_0i and P_1i, respectively, computed under the constraint p̃_1i = p̃_0i + PD.

Similarly as for unstratified analysis,the 2-sided 100(1 − α)% CI is given by the roots for PD = P₁ − P₀, and the bisection algorithm is used in the function to obtain the two roots (confidence interval) for PD.

p-value and Z-statistic

The Z-statistic is computed as:

$$Z_\text{diff}=\frac{\hat{p}_1^*-\hat{p}^*_0+S_0}{\sqrt{\sum_{i=1}^I(W_i/\sum_{k=1}^{K}W_k)^2\tilde{V}_i}}$$ where S₀ is pre-specified proportion difference under the null;

• p̃_0i and p̃_1i are MLE for P_0i and P_1i, respectively, computed under the constraint p̃_1i = p̃_0i + S₀.

The p-value can be calculated as stated above.