Fisher's Exact Test - Biostatistics

1. A researcher comparing two diagnostic tests found that Test A correctly identified 4 out of 5 positive cases, while Test B correctly identified 6 out of 9 positive cases. Both tests had no false positives in this small sample. When applying Fisher's exact test to compare the sensitivity of these tests, what is the key assumption that must hold for the test to be valid?
   Answer: The marginal totals representing the number of true positives in each test group are treated as fixed values Fisher's exact test is based on conditioning on the marginal totals, treating them as fixed. This means we assume the total number of positive cases for each test (5 for Test A, 9 for Test B) and the overall number of correctly identified cases are predetermined. Choice A describes a study design consideration but not a statistical assumption of Fisher's test. Choice C incorrectly focuses on modeling false negatives specifically. Choice D confuses power considerations with the fundamental assumptions required for test validity.
2. A clinical researcher tests whether a rare side effect occurs equally in two treatment groups. Group 1 (n=7) had 2 patients with the side effect, while Group 2 (n=5) had 0 patients with the side effect. In Fisher's exact test, how many different 2×2 tables share the same marginal totals as this observed data?
   Answer: 3 tables, with Group 1 side effects ranging from 0 to 2 patients When you encounter Fisher's exact test questions, focus on understanding how the test constrains the possible outcomes by fixing the marginal totals (row and column sums). In this problem, the observed 2×2 table has fixed margins: Group 1 has 7 patients total, Group 2 has 5 patients total, and exactly 2 patients across both groups experienced the side effect. Fisher's exact test asks: given these fixed totals, how many different ways could the data be arranged? Since the margins are fixed, if Group 1 has $k$ patients with the side effect, then Group 2 must have $(2-k)$ patients with the side effect. The constraint is that $k$ cannot exceed the smaller of: (1) Group 1's size (7), or (2) the total number of side effects (2). Therefore, $k$ can only be 0, 1, or 2, giving us exactly 3 possible tables. Answer A is correct. Looking at the wrong answers: B suggests 6 tables from "all possible combinations," but this ignores that Fisher's exact test fixes the marginal totals, severely limiting the possibilities. C proposes 8 tables based on distributing 2 side effects among 12 patients, but this calculation doesn't account for the group structure and fixed group sizes. D simply uses the total sample size (12) as the number of tables, which has no logical connection to how contingency tables work. Study tip: Remember that Fisher's exact test always fixes both row and column totals. Once you identify these constraints, the number of possible tables is determined by how the smallest total can be distributed while respecting all margins.
3. In a Fisher's exact test for a 2×2 contingency table, a researcher obtains a one-tailed p-value of 0.035. The study was designed to test whether Treatment X has a higher success rate than the control, but the data actually showed Treatment X had a lower success rate than expected. If the researcher wants to report a two-tailed test result instead, what should be the approach?
   Answer: Calculate probabilities for tables as extreme or more extreme in both directions, which may not equal exactly 2 × 0.035 For Fisher's exact test, the two-tailed p-value is calculated by summing probabilities of all tables as or more extreme than the observed table in either direction. Due to the discrete nature of the hypergeometric distribution, this sum may not equal exactly twice the one-tailed p-value. Choice A incorrectly assumes the relationship is always exactly 2:1. Choice C misunderstands that direction doesn't affect the two-tailed calculation. Choice D confuses significance levels with p-value calculation methods.
4. A researcher performs Fisher's exact test on a 2×2 table and finds that only 3 possible tables (including the observed one) have the same marginal totals. If the observed table has a probability of 0.15 under the null hypothesis, and the other two tables have probabilities of 0.05 and 0.30, what is the one-tailed p-value for the most extreme direction?
   Answer: 0.20, representing the sum of probabilities for tables as extreme or more extreme When you encounter Fisher's exact test questions, remember that this test calculates the exact probability of observing data as extreme or more extreme than what was actually seen, given fixed marginal totals. Fisher's exact test works by identifying all possible tables with the same marginal totals as your observed data, then calculating each table's probability under the null hypothesis. The p-value is the sum of probabilities for tables that are "as extreme or more extreme" in the direction of interest. In this problem, you have three possible tables with probabilities 0.15 (observed), 0.05, and 0.30. For a one-tailed test, you need to determine which direction represents "more extreme" than your observed table, then sum the probabilities accordingly. Since the question asks for "the most extreme direction," you're looking at the tail that includes the table with probability 0.05 (the smallest probability, hence most extreme). The one-tailed p-value would be 0.15 + 0.05 = 0.20, making choice A correct. Choice B (0.15) incorrectly gives only the probability of the exact observed table, not the cumulative probability for the tail. Choice C (0.50) mistakenly suggests this relates to the most likely configuration, but p-values measure extremeness, not likelihood. Choice D (0.05) only includes the most extreme table while ignoring the observed table's contribution to the tail. Remember: Fisher's exact test p-values always represent cumulative probabilities in the tail(s) of interest, never just single table probabilities. Always sum the probabilities for tables as extreme or more extreme than observed.
5. A researcher applies Fisher's exact test to determine if treatment response differs between males and females. The data shows 5 responding males out of 8 total males, and 1 responding female out of 4 total females. If the researcher reports a 'mid-p' value instead of the standard Fisher's exact p-value, what modification is being made to the calculation?
   Answer: Half the probability of the observed table is added to the sum of more extreme table probabilities When you encounter questions about Fisher's exact test modifications, focus on understanding how different p-value calculations handle the probability of the observed data itself. Fisher's exact test calculates p-values by summing probabilities of tables as extreme or more extreme than the observed table. The standard approach includes the full probability of the observed table in this sum. However, the mid-p modification makes a specific adjustment: it adds only half the probability of the observed table to the sum of probabilities from more extreme tables. This approach addresses the inherent conservatism of Fisher's exact test, which tends to produce p-values that are too large because it fully counts the observed outcome. Choice A correctly describes this modification - half the observed table's probability is added to the more extreme probabilities. Choice B incorrectly suggests using median probabilities across table configurations, which isn't how mid-p works. Choice C confuses the mid-p adjustment with multiple comparison corrections or two-tailed test adjustments - the mid-p method doesn't involve adjusting alpha levels. Choice D incorrectly references normal approximations, but mid-p still uses the exact hypergeometric distribution; it just modifies how the observed table's probability is incorporated. Study tip: Remember that "mid-p" literally means using the "middle" or half of the observed probability. When you see mid-p mentioned with any exact test, think "half the observed probability plus all the more extreme probabilities" - this creates less conservative p-values than the standard exact approach.
6. A researcher compares complication rates between two surgical procedures: Procedure A had 1 complication in 4 surgeries, Procedure B had 3 complications in 5 surgeries. For Fisher's exact test, the researcher must enumerate all possible 2×2 tables with the same marginal totals. What constraint determines the range of possible values for the 'Procedure A complications' cell?
   Answer: The value must be between max(0, total complications - Procedure B surgeries) and min(Procedure A surgeries, total complications) Fisher's exact test requires you to consider all possible 2×2 tables that could have occurred with the same marginal totals as your observed data. This means keeping the row totals (surgeries per procedure) and column totals (total complications vs. no complications) fixed while allowing the cell values to vary. From the given data: Procedure A has 4 surgeries, Procedure B has 5 surgeries, and there are 4 total complications (1+3) across both procedures. When enumerating possible tables, you need to determine how many complications Procedure A could theoretically have had. The correct range follows two logical constraints. The minimum possible complications for Procedure A is $\max(0, 4-5) = 0$ because you can't have negative complications, and if Procedure B had all possible complications, Procedure A would have none. The maximum is $\min(4, 4) = 4$ because Procedure A can't have more complications than either its total surgeries or the total complications observed. Choice B incorrectly focuses on sample sizes rather than the interaction between complications and surgeries. Choice C ignores the constraint that Procedure A can't exceed its own sample size of 4 surgeries. Choice D makes the fundamental error of using the observed data range (1-4) rather than considering all theoretically possible values given the marginal constraints. Remember: In Fisher's exact test, the key insight is that marginal totals are fixed, but internal cell values can vary within logical bounds determined by those margins. Always think about both the minimum and maximum constraints simultaneously.
7. A researcher wants to test the association between gender and disease status in a small cohort study. The data shows 4 diseased males out of 8 males total, and 1 diseased female out of 6 females total. Which probability distribution underlies the calculation of Fisher's exact test for this scenario?
   Answer: Hypergeometric distribution conditioning on the marginal totals of the contingency table When analyzing associations in small contingency tables, you need to understand which probability distribution governs the test statistic. Fisher's exact test is specifically designed for situations where sample sizes are too small for chi-square approximations to be reliable. Fisher's exact test calculates the probability of observing your data (or more extreme results) by treating the marginal totals as fixed. In your example, you condition on having exactly 8 males, 6 females, 5 total diseased individuals, and 9 total healthy individuals. Given these fixed margins, the test asks: "What's the probability of seeing exactly 4 diseased males?" This follows the hypergeometric distribution, which models sampling without replacement from a finite population with two categories. Choice A correctly identifies this hypergeometric distribution conditioning on marginal totals. Choice B is wrong because the binomial distribution assumes independent trials with constant probability, but Fisher's exact test fixes the total number of successes (diseased individuals), making trials dependent. Choice C confuses Fisher's exact test with Pearson's chi-square test - while chi-square tests do use the chi-square distribution, Fisher's exact test uses exact probabilities, not asymptotic approximations. Choice D is incorrect because Fisher's exact test doesn't rely on normal approximations at all; it calculates exact probabilities using discrete distributions. Study tip: Remember that Fisher's exact test = hypergeometric distribution with fixed margins. When you see "exact test" in small samples, think "hypergeometric conditioning on totals," not approximations or asymptotic distributions.
8. A researcher is comparing treatment outcomes between two small groups using Fisher's exact test. The contingency table shows 3 successes out of 8 patients in Group A and 7 successes out of 9 patients in Group B. What is the primary reason Fisher's exact test is more appropriate than the chi-square test for this analysis?
   Answer: The expected cell frequencies are too small for the chi-square approximation to be valid When choosing between Fisher's exact test and the chi-square test for analyzing contingency tables, you need to consider whether the assumptions for the chi-square approximation are met. The chi-square test relies on a normal approximation that requires adequate expected frequencies in each cell of the contingency table. Let's examine this data: Group A has 3/8 successes (5/8 failures) and Group B has 7/9 successes (2/9 failures). When you calculate the expected frequencies under the assumption of no association, several cells fall well below the conventional threshold of 5. The chi-square test's normal approximation becomes unreliable when expected frequencies are too small, potentially leading to inaccurate p-values. Fisher's exact test sidesteps this problem by calculating the exact probability distribution rather than relying on approximations, making it the appropriate choice here. Looking at the incorrect options: Option B is wrong because Fisher's exact test isn't inherently more conservative—it simply provides the exact p-value rather than an approximation. Option C misses the point entirely; unequal sample sizes don't determine test choice between these methods—many valid chi-square analyses have unequal groups. Option D reflects a misunderstanding of what Fisher's exact test does; it doesn't account for correlation between variables any differently than the chi-square test—both test for independence in contingency tables. Study tip: Always check expected cell frequencies when deciding between chi-square and Fisher's exact test. If any expected frequency is below 5 (some sources say below 1), or if more than 20% of cells have expected frequencies below 5, choose Fisher's exact test for small contingency tables.
9. A biostatistician is reviewing a colleague's analysis that used Fisher's exact test on a dataset with 50 patients per group and found a p-value of 0.048. The colleague argues that Fisher's exact test was necessary because 'it provides the exact p-value rather than an approximation.' What is the most appropriate critique of this reasoning?
   Answer: The 'exact' nature of Fisher's test refers to the probability calculation method, but both tests would yield very similar p-values with this sample size With large samples (50 per group), both Fisher's exact test and chi-square test would yield very similar p-values. The 'exact' in Fisher's exact test refers to using the exact hypergeometric distribution rather than asymptotic approximations, but this advantage becomes negligible with large samples. Choice A mentions computational intensity but doesn't address the core statistical reasoning. Choice B incorrectly characterizes Fisher's test as 'conservative' in large samples. Choice C is wrong - Fisher's exact test is valid for any sample size, though unnecessary for large samples.
10. A biostatistician performs Fisher's exact test on treatment data and obtains a one-tailed p-value of 0.028. The research question asks whether the treatments differ in either direction (non-directional alternative). If the distribution of possible tables under the null hypothesis is symmetric, what would be the two-tailed p-value?
    Answer: 0.056, assuming perfect symmetry allows doubling the one-tailed p-value When you encounter Fisher's exact test questions involving one-tailed versus two-tailed p-values, focus on understanding how the test distributes probability across possible outcomes under the null hypothesis. Fisher's exact test calculates the probability of observing your data or more extreme results, given the null hypothesis of no association. A one-tailed test examines extremeness in only one direction, while a two-tailed test considers both directions. When the distribution of possible tables under the null hypothesis is symmetric, the probability mass is evenly distributed on both sides. Since you obtained a one-tailed p-value of 0.028 and the distribution is symmetric, this represents the probability of observing results this extreme or more in one specific direction. For a non-directional (two-tailed) alternative hypothesis, you need to account for equally extreme results in the opposite direction. Under perfect symmetry, this opposite tail also contributes 0.028, making the two-tailed p-value $0.028 + 0.028 = 0.056$. Answer A correctly identifies this doubling relationship under symmetric conditions. Answer B incorrectly assumes the one-tailed p-value already accounts for both directions—it doesn't. Answer C mistakenly halves the p-value, confusing the conversion direction. Answer D is wrong because Fisher's exact test can absolutely be adapted for two-tailed hypotheses. Study tip: Remember that converting from one-tailed to two-tailed p-values under symmetry requires doubling, not halving. The key phrase "symmetric distribution" signals that this doubling relationship holds perfectly.
11. In a small randomized trial comparing two pain medications, 4 out of 6 patients improved with Drug A, while 1 out of 5 patients improved with Drug B. A statistician performs Fisher's exact test and finds three possible table configurations with the same marginal totals. If these tables have probabilities 0.12, 0.30, and 0.58 under the null hypothesis, which represents the observed data?
    Answer: Cannot be determined without knowing which table corresponds to the actual cell counts observed When you encounter Fisher's exact test questions, remember that this test evaluates all possible ways to arrange data while keeping the row and column totals (marginal totals) fixed. The key insight here is distinguishing between what the test calculates versus what actually happened in your study. The question describes a completed study with specific observed results: 4/6 patients improved with Drug A and 1/5 improved with Drug B. These are the actual data collected. Fisher's exact test then considers this observed configuration alongside all other possible arrangements that maintain the same marginal totals, calculating probabilities for each under the null hypothesis of no treatment difference. Answer A is correct because the question doesn't tell you which of the three probability values (0.12, 0.30, or 0.58) corresponds to the specific table containing 4 successes in the Drug A group and 1 success in the Drug B group. You know the observed data, but you don't know which probability matches those exact cell counts. Answer B incorrectly assumes the middle probability represents observed data. Answer C wrongly suggests the most extreme table (lowest probability) must be the observed one - but Fisher's exact test considers all possible tables regardless of which was actually observed. Answer D incorrectly identifies the most likely table under the null hypothesis as the observed data. Study tip: In Fisher's exact test problems, always distinguish between the observed data (what actually happened in your study) and the theoretical tables the test examines (all possible arrangements with the same margins).
12. A epidemiologist studies the association between exposure to a chemical (yes/no) and development of a rare disease (yes/no) in a small community of 12 people. The study finds 3 exposed individuals with disease, 2 exposed individuals without disease, 1 unexposed individual with disease, and 6 unexposed individuals without disease. What assumption of Fisher's exact test might be questionable in this scenario?
    Answer: The assumption that observations are independent may be violated due to shared community exposure sources When analyzing statistical tests like Fisher's exact test, you need to consider whether the underlying assumptions are met in your specific study context. Fisher's exact test assumes that observations are independent - meaning one person's exposure or disease status doesn't influence another's. In this community study, the independence assumption is highly questionable because all 12 people live in the same small community. They likely share environmental factors, water sources, air quality, and other exposure pathways that could affect both their chemical exposure and disease risk. If the chemical comes from a contaminated local well or factory, for example, exposure patterns aren't random but clustered by geography and shared resources. This violates the independence assumption that Fisher's exact test requires. Looking at the wrong answers: B is incorrect because Fisher's exact test actually works well for rare diseases and doesn't assume any particular disease prevalence. The binomial distribution assumption isn't problematic here. C is wrong because exposure (yes/no) and disease (yes/no) are clearly categorical variables, which is exactly what Fisher's exact test is designed for. D misunderstands the test - Fisher's exact test doesn't require approximately equal marginal totals. In fact, it's specifically useful when sample sizes are small and unbalanced, like in this 2×2 table with 5 exposed vs. 7 unexposed individuals. Study tip: When evaluating statistical test assumptions, always consider the real-world context of data collection. Geographic clustering, shared environments, or related individuals often violate independence assumptions, which is a common issue in epidemiological studies.
13. Given the scenario described above, when a biostatistician chooses Fisher's exact test over a chi-square test to analyze the treatment outcomes, what is the primary statistical justification for this decision?
    Answer: The limited sample size likely violates the minimum expected cell frequency requirements needed for chi-square test validity The primary justification is that with only 20 patients total, the expected cell frequencies in a 2×2 table are likely to be less than 5 in one or more cells, violating the chi-square test's requirement for adequate expected frequencies. Fisher's exact test doesn't rely on asymptotic approximations and is valid regardless of cell sizes. Choice A incorrectly suggests that having the entire population changes the statistical requirements. Choice B mischaracterizes the continuity correction issue. Choice D incorrectly implies Fisher's test is specific to rare conditions rather than small samples.
14. In a 2×2 contingency table analysis using Fisher's exact test, a researcher obtains a one-tailed p-value of 0.032. If the alternative hypothesis is non-directional, what would be the appropriate two-tailed p-value for this test?
    Answer: The two-tailed p-value could be 0.064, 0.032, or somewhere between, depending on the table symmetry When you encounter Fisher's exact test questions, remember that the relationship between one-tailed and two-tailed p-values isn't as straightforward as with symmetric distributions like the normal or t-distribution. Fisher's exact test calculates p-values by summing probabilities of all tables as extreme or more extreme than the observed table. For a one-tailed test, you sum probabilities in one direction. For a two-tailed test, you must sum probabilities in both directions—but here's the key insight: the distribution of possible table configurations is often asymmetric. The correct answer is A because the two-tailed p-value depends on the specific contingency table structure. If the table is perfectly symmetric, doubling gives you the exact two-tailed p-value ($2 \times 0.032 = 0.064$). However, most real tables are asymmetric, meaning extreme tables in one tail might not have corresponding tables of equal probability in the other tail. This results in a two-tailed p-value that could be 0.064, 0.032, or anywhere between. Option B incorrectly assumes perfect symmetry—this "doubling rule" only applies to symmetric distributions. Option C misunderstands the calculation entirely; a one-tailed result of 0.032 cannot simultaneously be the two-tailed result. Option D incorrectly reverses the relationship, suggesting you divide rather than account for both tails. Study tip: For Fisher's exact test, always remember that the exact two-tailed p-value requires examining the actual table structure. Unlike parametric tests, you can't simply double the one-tailed result due to the discrete, often asymmetric nature of contingency table probability distributions.
15. A pharmaceutical company conducted a clinical trial with 8 patients receiving Drug A and 7 patients receiving Drug B. Drug A showed 7 successes, Drug B showed 3 successes. Before performing Fisher's exact test, the statistician needs to determine the range of possible success counts for Drug A, given that the marginal totals are fixed. What is the minimum number of successes Drug A could have had while maintaining the same marginal totals?
    Answer: 3 successes, because this represents the scenario where Drug B gets all possible successes from the total available In Fisher's exact test, marginal totals are fixed. With 8 patients in Drug A group, 7 in Drug B group, and 10 total successes, the minimum for Drug A occurs when Drug B gets its maximum possible successes. Drug B can have at most 7 successes (limited by its group size), leaving a minimum of 10 - 7 = 3 successes for Drug A. Choice B incorrectly calculates 10 - 7 = 3 but states 4. Choice C ignores the constraint that Drug B can't exceed 7 successes. Choice D ignores that Drug B is limited to 7 patients maximum.
16. In Fisher's exact test, the null hypothesis of independence is tested by examining all possible 2×2 tables with the same marginal totals as the observed data. What key assumption about the data collection process makes this approach valid?
    Answer: The marginal totals represent fixed constraints from the study design rather than random outcomes Fisher's exact test is fundamentally based on the hypergeometric distribution, which models sampling without replacement from a finite population. Understanding when this approach is valid requires recognizing what makes the marginal totals "fixed" versus "random." The key insight is that Fisher's exact test assumes you're conditioning on the marginal totals - treating them as if they were predetermined by your study design rather than random outcomes. This happens in scenarios like case-control studies where you deliberately choose to sample a specific number of cases and controls, or in randomized experiments where you assign a fixed number of subjects to each treatment group. When marginal totals are truly fixed by design, the hypergeometric distribution correctly models all possible ways the data could have been distributed within those constraints, making answer A correct. Answer B describes a general assumption for most statistical tests but doesn't explain why we can condition on marginal totals specifically. Answer C refers to the assumption of constant probability, which is more relevant to binomial models than to the conditioning logic of Fisher's exact test. Answer D describes random sampling assumptions that apply broadly to inference but doesn't address the specific mathematical foundation that makes Fisher's exact test valid. Study tip: When you encounter Fisher's exact test questions, always ask yourself: "Were the row and column totals determined by the researcher's design choices, or did they emerge randomly from the data?" This distinction determines whether the hypergeometric conditioning approach is appropriate.
17. In a Fisher's exact test comparing treatment outcomes, a statistician calculates that 12 different 2×2 tables are possible with the given marginal totals. The observed table has a probability of 0.08 under the null hypothesis. For a two-tailed test with α = 0.05, which tables should be included in the p-value calculation?
    Answer: All tables with probability ≤ 0.08, regardless of which marginal cell they favor Fisher's exact test questions often trip up students because they require understanding how two-tailed p-values are calculated when dealing with discrete probability distributions rather than continuous ones. In Fisher's exact test, you calculate the p-value by summing probabilities of all tables that are "as extreme or more extreme" than your observed table. The key insight is that "extreme" means having a probability under the null hypothesis that's as small or smaller than what you observed, regardless of which direction the deviation goes. Since your observed table has probability 0.08, you include all possible tables with probability ≤ 0.08 in your p-value calculation. Answer A is correct because it captures this principle exactly - you sum probabilities of all tables with probability ≤ 0.08, whether they show effects in the same direction as your observed table or the opposite direction. Answer B misunderstands how the α level relates to table selection. You don't simply take the 5% most extreme tables; you take all tables as extreme as or more extreme than what you actually observed. Answer C describes a flawed hybrid approach that doesn't properly account for the discrete nature of the distribution. You can't just add "the single most extreme opposite table." Answer D incorrectly tries to split the discrete distribution into symmetric tails with exactly 0.025 probability each, which isn't possible with discrete data where only specific probability values exist. Remember: In Fisher's exact test, "two-tailed" means including all tables with probabilities as small or smaller than your observed table's probability, regardless of direction.
18. A clinical trial compares two treatments with the following results: Treatment A had 2 cures out of 5 patients, Treatment B had 0 cures out of 4 patients. When applying Fisher's exact test, what assumption is made about the marginal totals during the calculation?
    Answer: The marginal totals are treated as fixed constants given by the experimental design Fisher's exact test is used when you have small sample sizes in contingency tables and need to calculate exact p-values rather than relying on normal approximations. The key insight is understanding what assumptions the test makes about your data structure. Fisher's exact test treats the marginal totals (row and column sums) as fixed constants determined by your experimental design. In your example, you decided to treat 5 patients with Treatment A and 4 patients with Treatment B - these totals are fixed by design. Similarly, the total number of cures (2) is treated as fixed when calculating the exact probability distribution. The test then asks: "Given these fixed marginal totals, what's the probability of observing this particular arrangement or one more extreme?" Looking at the incorrect options: B) suggests marginal totals follow a binomial distribution, but Fisher's test doesn't model the totals as random - it conditions on them being fixed. C) claims marginal totals are random variables to be estimated, which contradicts Fisher's conditioning approach where totals are treated as known constants from your study design. D) assumes equal marginal totals between groups, but Fisher's test works perfectly fine with unequal group sizes (like your 5 vs 4 patients). The correct answer is A because Fisher's exact test fundamentally assumes marginal totals are fixed by your experimental design choices. Study tip: Remember that "exact" in Fisher's exact test refers to conditioning on fixed marginal totals - this is what allows the test to provide exact p-values for small samples without normal approximations.
19. A pharmaceutical company tests a new drug in a small trial with 6 patients receiving the drug (3 improved, 3 did not improve) and 4 patients receiving placebo (0 improved, 4 did not improve). When would Fisher's exact test be preferred over other tests of association for these data?
    Answer: When any expected cell frequency under independence would be less than 5 When analyzing categorical data in small samples, you need to decide between chi-square tests and Fisher's exact test based on whether the chi-square approximation is reliable. The key principle is that chi-square tests assume large sample sizes where expected frequencies follow a normal distribution. Fisher's exact test is preferred when the chi-square test's assumptions are violated, specifically when expected cell frequencies are too small. The standard rule is that if any expected cell frequency under the null hypothesis of independence is less than 5, you should use Fisher's exact test instead of chi-square. In this drug trial, with such small sample sizes (6 and 4 patients), calculating expected frequencies would likely yield values below 5 in several cells, making Fisher's exact test the appropriate choice. Looking at the incorrect options: Choice B is wrong because the decision isn't based solely on total sample size—it's about expected cell frequencies specifically. A study with 15 patients might still use chi-square if expected frequencies are adequate. Choice C is incorrect because having zero events in a category doesn't automatically require Fisher's exact test; it depends on whether expected frequencies meet the threshold. Choice D misses the point entirely—the choice between these tests is about statistical assumptions, not about directional versus non-directional hypotheses. Remember this rule: When you see small sample categorical data analysis questions, always think about expected cell frequencies first. If any expected frequency would be less than 5, Fisher's exact test is your go-to method for reliable results.
20. A researcher is planning to use Fisher's exact test to analyze data from a crossover study where each of 10 patients receives both treatments in random order, with a washout period between treatments. The outcome is binary (success/failure) for each treatment period. What is the fundamental issue with applying Fisher's exact test to this study design?
    Answer: The crossover design creates correlated observations within patients, violating the independence assumption required for Fisher's exact test Fisher's exact test, like other tests for independent groups, assumes that observations are independent. In a crossover design, each patient contributes data for both treatments, creating correlated observations within patients. This violates the fundamental independence assumption. The appropriate analysis would be McNemar's test for paired binary data. Choice B incorrectly suggests group size issues. Choice C mischaracterizes washout periods as creating missing data. Choice D incorrectly focuses on multiple comparisons rather than the correlation structure.

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