Hypothesis Testing: Two Independent Samples and Chi-Square Procedures

Chi-Square Goodness-of-Fit Test (One Sample, Categorical Outcome)

• Application: Used for a single sample with a categorical or ordinal outcome (with $k$ categories) to assess agreement between observed sample frequencies and frequencies expected under the null hypothesis.

• Hypotheses:

  • Null hypothesis: $H_0: p_1 = p_{10}, p_2 = p_{20}, \dots, p_k = p_{k0}$, where $p_{10}, p_{20}, \dots, p_{k0}$ represent the expected population proportions.
  • Alternative hypothesis: $H_A: H_0 \text{ is false}$.

• Expected Frequencies ($E$): Calculated as $E = n \times p_0$ for each category, where $n$ is the total sample size.

• Test Statistic:$\chi^2 = \sum \frac{(O - E)^2}{E}$

  • $O$ = observed frequency
  • $E$ = expected frequency under $H_0$
  • When $H_0$ is true, $\chi^2$ is close to $0$; when $H_0$ is false, $\chi^2$ is large.

• Degrees of Freedom & Decision Rule:

  • Degrees of freedom: $df = k - 1$
  • Decision rule: Reject $H_0$ if $\chi^2 > \text{Critical Value from Table 3}$.

Chi-Square Test of Independence (Two or More Samples, Categorical Outcome)

• Application: Used to evaluate whether two categorical or ordinal variables are related across two or more independent comparison groups.

• Hypotheses:

  • Null hypothesis: $H_0: \text{The distribution of the outcome is independent of the groups}$ (treatment/group and outcome are independent).
  • Alternative hypothesis: $H_A: H_0 \text{ is false}$.

• Expected Cell Frequency Calculation:$E = \frac{\text{Row Total} \times \text{Column Total}}{N}$

  • $N$ = total overall sample size.

• Test Statistic:$\chi^2 = \sum \frac{(O - E)^2}{E}$

• Degrees of Freedom & Decision Rule:

  • Degrees of freedom: $df = (r - 1)(c - 1)$, where $r$ is the number of rows and $c$ is the number of columns.
  • Decision rule: Reject $H_0$ if $\chi^2 > \text{Critical Value from Table 3}$.

Two Independent Samples: Continuous Outcome

• Application: Compares means ($\mu_1 - \mu_2$) between two independent groups for a continuous outcome variable (e.g., blood pressure, weight, cholesterol).

• Hypotheses:

  • Null hypothesis: $H_0: \mu_1 = \mu_2$ (or $\mu_1 - \mu_2 = 0$).
  • Alternative hypothesis options:
  • One-sided upper: $H_A: \mu_1 > \mu_2$
  • One-sided lower: $H_A: \mu_1 < \mu_2$
  • Two-sided: $H_A: \mu_1 \neq \mu_2$

• Equal Variance Assumption & Pooled Standard Deviation ($S_p$):

  • The equal variance assumption ($\sigma_1^2 = \sigma_2^2$) is considered reasonable if:     $0.5 < \frac{s_1^2}{s_2^2} < 2.0$
  • Pooled standard deviation formula:     $S_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

• Test Statistics by Sample Size:

  • Large samples ($n_1 > 30$ and $n_2 > 30$):     $Z = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
  • Small samples ($n_1 < 30$ or $n_2 < 30$):     $t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
  • Degrees of freedom: $df = n_1 + n_2 - 2$.

Two Independent Samples: Dichotomous Outcome

• Application: Compares proportions ($p_1 - p_2$) of a binary or dichotomous success/failure outcome across two independent groups.

• Hypotheses:

  • Null hypothesis: $H_0: p_1 = p_2$
  • Alternative hypothesis options:
  • One-sided upper: $H_A: p_1 > p_2$
  • One-sided lower: $H_A: p_1 < p_2$
  • Two-sided: $H_A: p_1 \neq p_2$

• Overall Pooled Proportion ($p$):$p = \frac{\text{Total Successes}}{\text{Total Sample Size}} = \frac{x_1 + x_2}{n_1 + n_2}$

• Test Statistics:

  • When sample sizes are large ($n_1 > 30$ and $n_2 > 30$):     $Z = \frac{p_1 - p_2}{\sqrt{p(1 - p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$
  • When sample sizes are small ($n_1 < 30$ or $n_2 < 30$):     $t = \frac{p_1 - p_2}{\sqrt{p(1 - p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$