Biostatistics Chapter 6: Population Proportions

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Questions and Answers

What is the null hypothesis when comparing two population proportions?

π1 ≠ π2

π1 > π2

π1 < π2

π1 = π2 (correct)

What value of z indicates the rejection of the null hypothesis in a one-sided test at the 0.05 significance level?

1.96

1.645

1.65 (correct)

1.5

In the context of McNemar’s chi-square test, when is the null hypothesis rejected in a two-sided test?

When χ2 ≥ 3.84 (correct)

When χ2 = 3.84

When χ2 > 3.84

When χ2 < 3.84

In the formula for calculating the z score, which component represents the sample proportion?

p Signup and view all the answers

What does the test statistic χ2 calculate in the context of paired data?

The standardized difference in proportions Signup and view all the answers

What are the sizes n1 and n2 in the context of comparing two independent samples?

The binary data sample sizes Signup and view all the answers

How is the z statistic formulated using the sample proportions?

z = (p - π0) / (π0 * (1 - π0) / n) Signup and view all the answers

What is the appropriate conclusion if the z-score calculated is 1.70 in a one-sided test?

Fail to reject the null hypothesis Signup and view all the answers

What is the mean of the sampling distribution of the sample proportion when testing a null hypothesis?

π0 Signup and view all the answers

What is the first step in performing a test of significance for H0?

Decide whether a one- or two-sided test is appropriate. Signup and view all the answers

In a two-sided test with a significance level of 0.05, what z-score threshold is used for the rejection region?

z ≤ -1.96 or z ≥ 1.96 Signup and view all the answers

What is the variance of the sampling distribution of the sample proportion?

π0(1 - π0) Signup and view all the answers

When would a one-sided test be appropriate?

When testing if π is greater than π0. Signup and view all the answers

What does the z score represent in the context of hypothesis testing?

The number of standard deviations the sample proportion is from the hypothesized value. Signup and view all the answers

In the analysis of pair-matched data, what is typically observed?

The same subject twice under different conditions. Signup and view all the answers

How is the z score calculated using sample proportion p and hypothesized proportion π0?

z = (p - π0) / sqrt(π0(1 - π0)/n) Signup and view all the answers

What significance level is commonly chosen when performing these tests?

0.05 Signup and view all the answers

Which formula is used to calculate the z score in comparing two proportions?

$ z = \frac{p_2 - p_1}{p \sqrt{p(1 - p)(\frac{1}{n_1} + \frac{1}{n_2})}} $ Signup and view all the answers

What is the rejection region for the one-sided alternative HA: π2 > π1 at α = 0.05?

z ≥ 1.65 Signup and view all the answers

What is the chi-square test statistic's threshold for rejecting the null hypothesis at the 0.05 level?

3.84 Signup and view all the answers

Which of the following formulas represents the chi-square statistic when data is in a 2 × 2 table?

$ \chi^2 = \frac{(a + b + c + d)(ad - bc)^2}{(a + c)(b + d)(a + b)(c + d)} $ Signup and view all the answers

What does the Mantel-Haenszel method help to investigate in biostatistics?

The relationship between two binary variables while controlling for confounders Signup and view all the answers

In performing a significance test, what is the purpose of referring to a standard normal distribution table?

To select the cut point for rejection regions depending on α Signup and view all the answers

What characterizes a pair-matched case-control study?

Individual cases are matched to controls based on confounding variables. Signup and view all the answers

In a 2 × 2 table for pair-matched data, what does 'b' represent?

Pairs where the case is exposed and the control is unexposed. Signup and view all the answers

Which scenario is least likely to provide evidence in a pair-matched case-control study?

Pairs with two unexposed members. Signup and view all the answers

What is the goal of analyzing pair-matched data?

To compare the incidence of exposure among cases versus controls. Signup and view all the answers

If a pair-matched study has n = b + c, what do 'b' and 'c' specifically represent?

The number of pairs where the case is exposed and control is unexposed (b) and unexposed case with exposed control (c). Signup and view all the answers

When conducting a pair-matched case-control study, what is a common method for selecting controls?

Taking disease-free individuals from the population at risk. Signup and view all the answers

Which of the following statements about pair-matched case-control studies is true?

Matching helps to control for confounding factors. Signup and view all the answers

What is represented by the symbol π0 in the analysis of pair-matched data?

The proportion of a particular binary outcome, often set to 0.5. Signup and view all the answers

What does the summation formula represent when assessing the relationship between disease and exposure?

A pooled odds ratio considering the confounding variable Signup and view all the answers

In a chi-square test for one degree of freedom, what indicates that the association between disease and exposure is real?

The test is statistically significant Signup and view all the answers

What is the formula to estimate the odds ratio at each confounder level?

ad/bc Signup and view all the answers

What does the Mantel-Haenszel procedure accomplish in statistical analysis?

It pools data across levels of a confounder for a combined estimate Signup and view all the answers

What is represented by πij in the context of a two-way table?

The joint probability of outcomes from both categorical variables Signup and view all the answers

What occurs when two categorical variables in a two-way table are independent?

The joint probabilities equal the product of the marginal probabilities Signup and view all the answers

Under the assumption that the confounder is not an effect modifier, what can be concluded about the odds ratio?

It is a fixed value across all levels of the confounder Signup and view all the answers

What does the notation ∑(ad/n) represent in the Mantel-Haenszel method?

The sum of individual odds ratios weighted by sample size Signup and view all the answers

What do the eij values represent in a contingency table analysis?

Estimated expected frequencies under the null hypothesis of independence Signup and view all the answers

What is the primary goal of performing a test for independence?

To see if the two factors or variables X1 and X2 are related Signup and view all the answers

What does Pearson's chi-square statistic measure in a contingency table?

The discrepancy between observed and expected frequencies Signup and view all the answers

Which condition must be satisfied for Pearson's chi-square test to be appropriate?

Each expected frequency must be at least 5 Signup and view all the answers

What occurs if the calculated chi-square statistic is greater than the critical value?

The null hypothesis of independence is rejected Signup and view all the answers

What is Fisher's Exact Test primarily used for?

To analyze relationships in small samples Signup and view all the answers

What represents the degrees of freedom (df) in the context of a chi-square test for independence?

$df = (I - 1)(J - 1)$ Signup and view all the answers

What is the central idea behind Fisher's Exact Test?

To enumerate all possible outcomes with calculated probabilities Signup and view all the answers

Study Notes

Comparison of Population Proportions

This presentation covers methods for comparing population proportions, including one-sample problems, analysis of paired data, comparison of two proportions, and more complex cases like Mantel-Haenszel method, Fisher's exact test, and ordered 2xK contingency tables.
It draws from Le and Eberly (2016) "Introductory Biostatistics" 2nd edition.

Content Outline

6.1 One-Sample Problem with Binary Data: Deals with a sample of binary data (n, x) where n is the sample size and x is the number of positive outcomes. Null hypothesis (H₀) is π = π₀ (a fixed, known number between 0 and 1). The problem is testing if the observed proportion differs from a standardized or referenced figure.
6.2 Analysis of Pair-Matched Data: Suitable when each subject or member of a group is observed twice or pairs are observed for the same characteristic (e.g., hospital admissions, matched pairs). Common application is case-control studies (cases of specific diseases are compared to controls). The goal is to compare the exposure incidence.
6.3 Comparison of Two Proportions: This involves two independent samples of binary data (n₁, x₁) and (n₂, x₂). The sample sizes (n₁) and (n₂) may be equal or different; the x values represent the number of positive outcomes in each sample. The null hypothesis (H₀) is π₁ = π₂ (equality of the proportions). Steps include choosing a one-tailed or two-tailed test and calculating a z-score based on a pooled proportion.
6.4 Mantel-Haenszel Method: An approach for investigating the relationship between two binary variables (e.g., disease and exposure) while controlling for a confounder (a variable associated with either the disease or exposure or both).
6.5 Inferences for General Two-Way Tables: Examines the general case of an I x J table (resulting from a survey of size n) where X₁ and X₂ are two categorical variables with I and J levels, respectively. This explores the IJ combinations of classifications and the probabilities. It identifies the probability πij that the outcome falls into cell (i, j) in a two-way table. The analysis considers independence between two categorical variables.
6.6 Fisher's Exact Test: A method used for small sample sizes to determine if there's a statistically significant association between variables in a 2x2 contingency table, where the expected frequencies may be low according to the guidelines.
6.7 Ordered 2 x K Contingency Tables: This section deals with more complex, ordered 2 x k tables analyzing, for example, the concordance and discordance between categorical variables.

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Description

This quiz covers key methods for comparing population proportions as discussed in Chapter 6 of 'Introductory Biostatistics' by Le and Eberly. Learn about one-sample problems, paired data, and complex cases like Fisher's exact test. Test your understanding of these essential biostatistical concepts.