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>p (D)</p> Signup and view all the answers

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

<p>The standardized difference in proportions (A)</p> Signup and view all the answers

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

<p>The binary data sample sizes (B)</p> Signup and view all the answers

How is the z statistic formulated using the sample proportions?

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

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

<p>Fail to reject the null hypothesis (B)</p> Signup and view all the answers

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

<p>π0 (A)</p> Signup and view all the answers

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

<p>Decide whether a one- or two-sided test is appropriate. (C)</p> 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?

<p>z ≤ -1.96 or z ≥ 1.96 (C)</p> Signup and view all the answers

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

<p>π0(1 - π0) (C)</p> Signup and view all the answers

When would a one-sided test be appropriate?

<p>When testing if π is greater than π0. (B)</p> Signup and view all the answers

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

<p>The number of standard deviations the sample proportion is from the hypothesized value. (A)</p> Signup and view all the answers

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

<p>The same subject twice under different conditions. (B)</p> Signup and view all the answers

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

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

What significance level is commonly chosen when performing these tests?

<p>0.05 (D)</p> Signup and view all the answers

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

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

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

<p>z ≥ 1.65 (D)</p> 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?

<p>3.84 (A)</p> Signup and view all the answers

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

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

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

<p>The relationship between two binary variables while controlling for confounders (D)</p> Signup and view all the answers

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

<p>To select the cut point for rejection regions depending on α (C)</p> Signup and view all the answers

What characterizes a pair-matched case-control study?

<p>Individual cases are matched to controls based on confounding variables. (C)</p> Signup and view all the answers

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

<p>Pairs where the case is exposed and the control is unexposed. (A)</p> Signup and view all the answers

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

<p>Pairs with two unexposed members. (A), Pairs with two exposed members. (C)</p> Signup and view all the answers

What is the goal of analyzing pair-matched data?

<p>To compare the incidence of exposure among cases versus controls. (D)</p> Signup and view all the answers

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

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

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

<p>Taking disease-free individuals from the population at risk. (C)</p> Signup and view all the answers

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

<p>Matching helps to control for confounding factors. (A)</p> Signup and view all the answers

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

<p>The proportion of a particular binary outcome, often set to 0.5. (B)</p> Signup and view all the answers

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

<p>A pooled odds ratio considering the confounding variable (D)</p> 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?

<p>The test is statistically significant (D)</p> Signup and view all the answers

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

<p>ad/bc (B)</p> Signup and view all the answers

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

<p>It pools data across levels of a confounder for a combined estimate (C)</p> Signup and view all the answers

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

<p>The joint probability of outcomes from both categorical variables (C)</p> Signup and view all the answers

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

<p>The joint probabilities equal the product of the marginal probabilities (B)</p> 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?

<p>It is a fixed value across all levels of the confounder (D)</p> Signup and view all the answers

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

<p>The sum of individual odds ratios weighted by sample size (D)</p> Signup and view all the answers

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

<p>Estimated expected frequencies under the null hypothesis of independence (A)</p> Signup and view all the answers

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

<p>To see if the two factors or variables X1 and X2 are related (A)</p> Signup and view all the answers

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

<p>The discrepancy between observed and expected frequencies (B)</p> Signup and view all the answers

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

<p>Each expected frequency must be at least 5 (A)</p> Signup and view all the answers

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

<p>The null hypothesis of independence is rejected (C)</p> Signup and view all the answers

What is Fisher's Exact Test primarily used for?

<p>To analyze relationships in small samples (B)</p> Signup and view all the answers

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

<p>$df = (I - 1)(J - 1)$ (C)</p> Signup and view all the answers

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

<p>To enumerate all possible outcomes with calculated probabilities (B)</p> Signup and view all the answers

Flashcards

One-sample problem with binary data

Statistical test for a single population proportion (π).

Null Hypothesis (H0)

The hypothesis that there is no significant difference between the sample proportion (p) and the population proportion (π0).

Sample Proportion (p)

Proportion of subjects or members with a particular characteristic in a sample.

Population Proportion (π0)

The assumed or hypothesized proportion of the population with a certain characteristic.

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Standard Error, σp

Standard deviation of the sample proportion (p)

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Z-score calculation

Standardized value representing how many standard errors a sample proportion is away from the hypothesized population proportion.

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Rejection Region

Values of the test statistic (z-score) that lead to rejecting the null hypothesis.

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Pair-matched data analysis

Statistical method for comparing outcomes between subjects or paired observations.

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Pair-matched case-control study

An epidemiological design where cases with a specific disease are matched to controls with similar characteristics (e.g., age, gender).

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Case-control study

A type of observational study where researchers compare individuals with a disease (cases) to individuals without the disease (controls).

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Control

An individual in a study who does not have the disease or condition being investigated; matched with a case.

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Confounding Variable

Variables that can influence both the exposure and the outcome, potentially creating a false association.

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Exposure

The factor or characteristic being investigated for its possible role in causing the disease.

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2x2 table

A table used for analyzing pair-matched data, showing exposure status (exposed/not exposed) in cases and their matched controls.

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One-sample problem

Statistical methodology for analyzing data with cases and controls in matched pairs.

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McNemar's chi-square test

A statistical test used to analyze paired binary data, comparing proportions between two related groups.

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Test Statistic (McNemar's)

Calculated as (b-c)^2 / (b+c), where b and c represent counts in a specific contingency table.

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One-sided test (McNemar's)

A hypothesis test where the alternative hypothesis specifies a direction for change (e.g., only an increase or only a decrease).

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Two-sided test (McNemar's)

A statistical test that investigates whether there is a difference between two groups without a specific direction of change.

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Independent samples

Two samples of binary data (e.g., success or failure) that are not related or matched.

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Null Hypothesis (Two Proportions)

The hypothesis that the proportions of success in two independent groups are equal.

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Population Proportions

The fraction of individuals in a population exhibiting a particular characteristic.

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Comparison of Two Proportions

Statistical analysis comparing the proportions of a particular characteristic between two independent groups.

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Significance test for Proportions

A statistical test to determine if the difference between two population proportions is statistically significant.

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One-sided vs. Two-sided test

Choosing a one-sided test (e.g., π2 > π1) specifies a direction of the difference, while a two-sided test (e.g., π1 ≠ π2) explores any difference.

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Significance level (α)

The probability of rejecting the null hypothesis when it is actually true. Commonly set to 0.05.

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Pooled Proportion

An estimate of the common proportion under the null hypothesis (no difference between groups). Calculated as (x1 + x2) / (n1 + n2).

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Chi-square test

Alternative method for testing the difference between two proportions, calculated from the contingency table. Useful when dealing with proportions.

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Rejection Region (z-score)

The range of z-scores that lead to rejection of the null hypothesis. Determined by the significance level and test type (one-sided vs. two-sided).

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Mantel-Haenszel method

A statistical method used to analyze the relationship between two binary variables while controlling for potential confounding variables.

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Odds Ratio (OR)

The ratio of odds of an event in one group to the odds of the event in another group. It quantifies the association between an exposure and an outcome.

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Confounder

A variable that is associated with both the exposure and the outcome, but is not on the causal pathway between them. It can distort the apparent association between exposure and outcome.

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Combined Odds Ratio (ORMH)

The overall estimate of the odds ratio obtained using the Mantel-Haenszel method, taking into account the influence of the confounder.

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How does the Mantel-Haenszel method control for confounding?

The Mantel-Haenszel method controls for confounding by calculating a weighted average of the odds ratios at each level of the confounder. This weighted average gives a more accurate estimate of the true association between exposure and outcome, free from the influence of the confounder.

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What is the assumption about the confounder in the Mantel-Haenszel method?

The Mantel-Haenszel method assumes that the confounder is not an effect modifier, meaning the odds ratio stays constant across different levels of the confounder. In other words, the strength of the association between exposure and outcome doesn't change depending on the level of the confounder.

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What does a statistically significant Mantel-Haenszel test indicate?

A statistically significant Mantel-Haenszel test indicates that the observed association between the exposure and the outcome is unlikely due to random chance, suggesting a real association between the two variables after controlling for the confounder.

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How does the Mantel-Haenszel method apply to a general two-way table with categorical variables?

The Mantel-Haenszel method can be applied to any two-way table with categorical variables, not just those with a binary outcome. It can be used to estimate the overall association between two categorical variables while controlling for confounding variables.

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Expected Frequencies

The frequencies we expect to see in each cell of a contingency table if the two variables are independent. They are calculated based on the marginal totals of the observed data.

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Chi-Square Statistic

A statistical test used to compare the observed frequencies in a contingency table to the expected frequencies under the null hypothesis of independence.

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Degrees of Freedom (df)

The number of independent values that can vary in a contingency table. It is calculated as (rows - 1) * (columns - 1).

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Rejection of Null Hypothesis

When the calculated chi-square statistic is greater than a critical value, we reject the null hypothesis of independence, suggesting that there is a relationship between the two variables.

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Fisher's Exact Test

A statistical test used to analyze contingency tables with small sample sizes, where the expected frequencies are less than 5. It calculates the exact probability of observing the observed frequencies or more extreme ones, assuming independence.

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What does Fisher's Exact Test calculate?

It calculates the exact probability of observing the observed frequencies or more extreme ones in a contingency table, assuming independence between the variables.

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When is Fisher's Exact Test preferred?

It is preferred for contingency tables with small sample sizes, specifically when the expected frequencies in any cell are less than 5.

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Why are small expected frequencies problematic?

Small expected frequencies can lead to inaccurate results with the chi-square test, as the distribution is not an accurate approximation.

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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.

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