ISTA 116 Hypothesis Testing: Binary Data
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1 ISTA 116 Hypothesis Testing: Binary Data November 14, 2013
2 Types of Errors Example H 1 : Drug is better than a placebo H 0 : Drug no better than a placebo We reject H 0 if the data would be improbable on the assumption that H 0 is true.
3 Types of Errors Example H 1 : Drug is better than a placebo H 0 : Drug no better than a placebo We reject H 0 if the data would be improbable on the assumption that H 0 is true. But improbable things happen sometimes! This means that we will occasionally reject H 0 incorrectly!
4 Types of Errors Example H 1 : Drug is better than a placebo H 0 : Drug no better than a placebo We reject H 0 if the data would be improbable on the assumption that H 0 is true. But improbable things happen sometimes! This means that we will occasionally reject H 0 incorrectly! E.g., we conclude that the drug works when in fact it doesn t: reject H 0 by mistake.
5 Types of Errors Example H 1 : Drug is better than a placebo H 0 : Drug no better than a placebo We could prevent this from ever happening by never rejecting H 0
6 Types of Errors Example H 1 : Drug is better than a placebo H 0 : Drug no better than a placebo We could prevent this from ever happening by never rejecting H 0 But then we make it more likely that we make the opposite error (e.g., fail to discover valuable new drugs)
7 Types of Errors We can summarize the possibilities in a contingency table, where one dimension is whether H 0 or H 1 is actually correct (does the student actually know stuff?), and the other is whether or not H 0 is rejected (do we conclude that he knows stuff?). Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) False Negative H 0 is true False Positive No Error Table: Possible outcomes of a null hypothesis significance test
8 Types of Errors False positives (H 0 incorrectly rejected and H 1 endorsed) and false negatives (H 0 is incorrectly retained and H 1 rejected) are called Type I Errors and Type II Errors, respectively. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
9 Error Rates The Type I Error Rate is the probability that we make a Type I Error out of the times that H 0 is true. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
10 Error Rates The Type I Error Rate is the probability that we make a Type I Error out of the times that H 0 is true. The Type II Error Rate is the probability that we make a Type II Error out of the times that H 0 is false. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
11 Error Rates The Type I Error Rate is the probability that we make a Type I Error out of the times that H 0 is true. The Type II Error Rate is the probability that we make a Type II Error out of the times that H 0 is false. These are probabilities. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
12 Error Rates The Type I Error Rate is the probability that we make a Type I Error out of the times that H 0 is true. The Type II Error Rate is the probability that we make a Type II Error out of the times that H 0 is false. These are conditional probabilities. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
13 Error Rates Type I Error Rate: P(Reject H 0 H 0 true) Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
14 Error Rates Type I Error Rate: P(Reject H 0 H 0 true) Type II Error Rate: P(Not Reject H 0 H 0 false) Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
15 Choosing a Significance Level Recall, we want to reject H 0 when the data would have been unlikely if H 0 were true. Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
16 Choosing a Significance Level Recall, we want to reject H 0 when the data would have been unlikely if H 0 were true. How unlikely is unlikely? We have a choice, as long as we set the threshold before we collect any data. We call this threshold α (it represents a probability). Action H 0 rejected H 0 not rejected Truth H 0 is false No Error (Hit) Type II Error H 0 is true Type I Error No Error Table: Possible outcomes of a null hypothesis significance test
17 Choosing a Significance Level α is the Type I Error rate: the conditional proportion of the time that we expect to incorrectly reject H 0 when it is true. Action H 0 rejected H 0 not rejected H 0 is false 1 β β 1 H 0 is true α 1 α 1 Table: Conditional Error Probabilities associated with a NHST
18 Choosing a Significance Level α is the Type I Error rate: the conditional proportion of the time that we expect to incorrectly reject H 0 when it is true. We call the Type II Error rate β. This is the conditional proportion of the time that we expect to fail to reject H 0 when we should have. Action H 0 rejected H 0 not rejected H 0 is false 1 β β 1 H 0 is true α 1 α 1 Table: Conditional Error Probabilities associated with a NHST
19 Choosing a Significance Level α is the Type I Error rate: the conditional proportion of the time that we expect to incorrectly reject H 0 when it is true. We call the Type II Error rate β. This is the conditional proportion of the time that we expect to fail to reject H 0 when we should have. The quantity 1 β tells us the conditional probability that we will be able to endorse our initial hypothesis (H 1 ) when it is true. This is also called the power of the test. Action H 0 rejected H 0 not rejected H 0 is false 1 β β 1 H 0 is true α 1 α 1 Table: Conditional Error Probabilities associated with a NHST
20 Step 3: Establish a Rejection Criterion How many does the student need to get correct for us to reject H 0? Probability H Values
21 Step 3: Establish a Rejection Criterion Enough so that we won t have too many Type I Errors (false positives). Probability H Values
22 Step 3: Establish a Rejection Criterion Convention: limit α to 5%. If we set the threshold to the 95th percentile of the H 0 distribution, it will only be exceeded 5% of the time that H 0 is true. Probability H Values
23 Step 3: Establish a Rejection Criterion P(X x) x How high does c have to be so that P(X c H 0 ) is less than 5%?
24 Step 3: Establish a Rejection Criterion P(X x) x How high does c have to be so that P(X c H 0 ) is less than 5%? If we reject H 0 when X 9, we will make a Type I Error less than 5% of the time.
25 Step 4: Get the data, and check! Suppose the student gets 8 correct. Then we would...
26 Step 4: Get the data, and check! Suppose the student gets 8 correct. Then we would... Our rejection criterion was 9 and above, so we cannot reject H 0 in this case.
27 Type I vs. Type II Errors We can set α to whatever we want. The lower it is, the less often we make Type I Errors.
28 Type I vs. Type II Errors We can set α to whatever we want. The lower it is, the less often we make Type I Errors. So why not make it really small?
29 Type I vs. Type II Errors We can set α to whatever we want. The lower it is, the less often we make Type I Errors. So why not make it really small? Tradeoff: Fewer Type I Errors More Type II Errors.
30 Type I vs. Type II Errors Decreasing α moves the threshold out toward the tail of the H 0 distribution. Probability α = 0.15, c = Values
31 Type I vs. Type II Errors Decreasing α moves the threshold out toward the tail of the H 0 distribution. Probability α = 0.05, c = Values
32 Type I vs. Type II Errors Decreasing α moves the threshold out toward the tail of the H 0 distribution. Probability α = 0.01, c = Values
33 Type I vs. Type II Errors We will retain H 0 when we do not exceed the threshold. But if H 1 is correct, this is a Type II Error. The more stringent the threshold, the more likely Type II Errors become (and so we miss new discoveries). Probability α = 0.15, c = Values
34 Type I vs. Type II Errors We will retain H 0 when we do not exceed the threshold. But if H 1 is correct, this is a Type II Error. The more stringent the threshold, the more likely Type II Errors become (and so we miss new discoveries). Probability α = 0.05, c = Values
35 Type I vs. Type II Errors We will retain H 0 when we do not exceed the threshold. But if H 1 is correct, this is a Type II Error. The more stringent the threshold, the more likely Type II Errors become (and so we miss new discoveries). Probability α = 0.01, c = Values
36 Hypothesis Testing Procedure 1. Translate research question into quantitative null and alternative hypotheses (H 0 and H 1 ) about the population/long run. 2. Identify a test statistic (what quantity (random variable) are we going to use to summarize the data?) 3. Establish a rejection criterion. (a) Find the distribution that the test statistic would have if H 0 (or the version of H 0 closest to H 1 ) is true. (b) Decide which data values would favor H 1 over H 0. (c) Out of these, set a cutoff so that the (conditional) probability of a Type I Error rate is low enough (usually 5% or less). 4. Collect the data, compute the test statistic, and make the decision.
37 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 more often than it would if the die were fair.
38 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 more often than it would if the die were fair. How could we test this?
39 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 more often than it would if the die were fair. How could we test this? We set up an experiment where we will roll the die 300 times.
40 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 more often than it would if the die were fair. How could we test this? We set up an experiment where we will roll the die 300 times. What are our null and alternative hypotheses?
41 Testing the Fairness of a Die Relevant aspect of the die is
42 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair
43 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased toward 1
44 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased toward 1 p 1 > 1/6 Relevant quantity from the data (test statistic) is
45 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased toward 1 p 1 > 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a distribution.
46 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased toward 1 p 1 > 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a B(300, 1/6) distribution. A high number of 1s favor H 1. How high?
47 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased toward 1 p 1 > 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a B(300, 1/6) distribution. A high number of 1s favor H 1. How high? We should reject H 0 if we get a value which is above the 95th percentile of the B(300, 1/6) distribution.
48 Testing the Fairness of a Die Notice that we don t actually need to find the exact cutoff in advance. If we get an observed test statistic, we can just compute its percentile in the H 0 distribution.
49 Testing the Fairness of a Die Notice that we don t actually need to find the exact cutoff in advance. If we get an observed test statistic, we can just compute its percentile in the H 0 distribution. If this percentile is, then we can reject H 0.
50 Testing the Fairness of a Die Notice that we don t actually need to find the exact cutoff in advance. If we get an observed test statistic, we can just compute its percentile in the H 0 distribution. If this percentile is greater than 95, then we can reject H 0.
51 Testing the Fairness of a Die Suppose we get 60 1 s. How do we proceed?
52 Testing the Fairness of a Die Suppose we get 60 1 s. How do we proceed? We need to determine the probability of 60 or a value more supportive of H 1 in the B(300, 1/6) distribution.
53 Testing the Fairness of a Die Suppose we get 60 1 s. How do we proceed? We need to determine the probability of 60 or a value more supportive of H 1 in the B(300, 1/6) distribution. High values are more supportive in this case, so we want P(X 60).
54 Testing the Fairness of a Die Suppose we get 60 1 s. How do we proceed? We need to determine the probability of 60 or a value more supportive of H 1 in the B(300, 1/6) distribution. High values are more supportive in this case, so we want P(X 60).
55 Testing the Fairness of a Die Suppose we get 60 1 s. How do we proceed? We need to determine the probability of 60 or a value more supportive of H 1 in the B(300, 1/6) distribution. High values are more supportive in this case, so we want P(X 60). 1 - pbinom(59, 300, 1/6) [1] What s our decision? We cannot reject H 0, because we get values this high more than 5% of the time under H 0.
56 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 less often than it would if the die were fair.
57 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 less often than it would if the die were fair. How could we test this?
58 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 less often than it would if the die were fair. How could we test this? We set up an experiment where we will roll the die 300 times.
59 Testing the Fairness of a Die Suppose we suspect that a die is loaded to come up 1 less often than it would if the die were fair. How could we test this? We set up an experiment where we will roll the die 300 times. What are our null and alternative hypotheses?
60 Testing the Fairness of a Die Relevant aspect of the die is
61 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair
62 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased away from 1
63 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased away from 1 p 1 < 1/6 Relevant quantity from the data (test statistic) is
64 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased away from 1 p 1 < 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a distribution.
65 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased away from 1 p 1 < 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a B(300, 1/6) distribution. A low number of 1s favor H 1. How low?
66 Testing the Fairness of a Die Relevant aspect of the die is the probability of a 1. Call this p 1. H 0 : Die is fair p 1 = 1/6 H 1 : Die is biased away from 1 p 1 < 1/6 Relevant quantity from the data (test statistic) is the number of 1s rolled If H 0 is true, this test statistic has a B(300, 1/6) distribution. A low number of 1s favor H 1. How low? We should reject H 0 if we get a value which is below the 5th percentile of the B(300, 1/6) distribution.
67 R tip: the qbinom() function The pbinom(q, size, prob) function gives us the value of the CDF at the value q, in a binomial distribution with parameters n = size and p = prob.
68 R tip: the qbinom() function The pbinom(q, size, prob) function gives us the value of the CDF at the value q, in a binomial distribution with parameters n = size and p = prob. What if we want the opposite? That is, we want to know what value is at the 5th percentile?
69 R tip: the qbinom() function The pbinom(q, size, prob) function gives us the value of the CDF at the value q, in a binomial distribution with parameters n = size and p = prob. What if we want the opposite? That is, we want to know what value is at the 5th percentile? We can use qbinom(p, size, prob).
70 R tip: the qbinom() function The pbinom(q, size, prob) function gives us the value of the CDF at the value q, in a binomial distribution with parameters n = size and p = prob. What if we want the opposite? That is, we want to know what value is at the 5th percentile? We can use qbinom(p, size, prob).
71 R tip: the qbinom() function The pbinom(q, size, prob) function gives us the value of the CDF at the value q, in a binomial distribution with parameters n = size and p = prob. What if we want the opposite? That is, we want to know what value is at the 5th percentile? We can use qbinom(p, size, prob). (x <- qbinom(0.05, 300, 1/6)) [1] 40
72 R tip: the qbinom() function Did it work? If we plug this value back in to pbinom(), will we get 0.05?
73 R tip: the qbinom() function Did it work? If we plug this value back in to pbinom(), will we get 0.05?
74 R tip: the qbinom() function Did it work? If we plug this value back in to pbinom(), will we get 0.05? pbinom(x, 300, 1/6) [1]
75 R tip: the qbinom() function Did it work? If we plug this value back in to pbinom(), will we get 0.05? pbinom(x, 300, 1/6) [1] What went wrong?
76 R tip: the qbinom() function Here s the inverse CDF of the B(300, 1/6) distribution. x P(X x)
77 R tip: the qbinom() function Let s zoom in on the bottom 10% x P(X x)
78 Avoiding Probability Creep Everything from the 4.86 percentile to the 6.75 percentile falls in the 40 bin. If we want to keep the Type I Error rate below 5%, we should reject H 0 only when the observed value is strictly below the 5th percentile. Similarly, when we re looking for high values, we should reject only if the observed value is strictly above the 95th percentile.
79 Outline Two-Tailed Tests
80 Non-directional Hypotheses We could similarly test a coin for bias. Suppose rather than having a specific hypothesis about the direction of bias, we just want to test for bias generally.
81 Non-directional Hypotheses We could similarly test a coin for bias. Suppose rather than having a specific hypothesis about the direction of bias, we just want to test for bias generally. Remember that we are willing to make a Type I Error α (say 0.05) of the time that the null hypothesis is true.
82 Non-directional Hypotheses We could similarly test a coin for bias. Suppose rather than having a specific hypothesis about the direction of bias, we just want to test for bias generally. Remember that we are willing to make a Type I Error α (say 0.05) of the time that the null hypothesis is true. We flip the coin 500 times, and get heads 230 times. What do we want to calculate?
83 Non-directional Hypotheses We could similarly test a coin for bias. Suppose rather than having a specific hypothesis about the direction of bias, we just want to test for bias generally. Remember that we are willing to make a Type I Error α (say 0.05) of the time that the null hypothesis is true. We flip the coin 500 times, and get heads 230 times. What do we want to calculate? Proceeding as before, we can represent the number of heads under the null hypothesis using a binomial distribution with n = 500 and p = 0.5.
84 Non-directional Hypotheses We could similarly test a coin for bias. Suppose rather than having a specific hypothesis about the direction of bias, we just want to test for bias generally. Remember that we are willing to make a Type I Error α (say 0.05) of the time that the null hypothesis is true. We flip the coin 500 times, and get heads 230 times. What do we want to calculate? Proceeding as before, we can represent the number of heads under the null hypothesis using a binomial distribution with n = 500 and p = 0.5. In this case, we are interested in P(Y 230). Using R, we can do pbinom(230, 500, 0.5), which gives us about What do we conclude?
85 Here s the spike plot Probability P(X 230) Values
86 Non-directional Hypotheses Suppose instead we had gotten 270 heads (i.e., 230 tails). What would we do?
87 Non-directional Hypotheses Suppose instead we had gotten 270 heads (i.e., 230 tails). What would we do? Now we want to know P(Y 270), which is
88 Non-directional Hypotheses Suppose instead we had gotten 270 heads (i.e., 230 tails). What would we do? Now we want to know P(Y 270), which is also 0.04, by symmetry
89 Non-directional Hypotheses Probability P(X 270) Values
90 Non-directional Hypotheses Here s both sides at once. Probability P(X 230) 0.04 P(X 270) Values
91 Non-directional Hypotheses So 4% of the time that we flip a fair coin 500 times we get 230 or fewer heads, and 4% of the time we get 270 or more heads. If the coin actually is fair and we get 230 or fewer, or 270 or more, we will reject H 0, which will be a.
92 Non-directional Hypotheses So 4% of the time that we flip a fair coin 500 times we get 230 or fewer heads, and 4% of the time we get 270 or more heads. If the coin actually is fair and we get 230 or fewer, or 270 or more, we will reject H 0, which will be a Type I Error. How often will we make Type I Errors of this sort?
93 Non-directional Hypotheses So 4% of the time that we flip a fair coin 500 times we get 230 or fewer heads, and 4% of the time we get 270 or more heads. If the coin actually is fair and we get 230 or fewer, or 270 or more, we will reject H 0, which will be a Type I Error. How often will we make Type I Errors of this sort? 8% in total! But wait, we re supposed to make Type I Errors only 5% of the time. What went wrong?
94 Non-directional Hypotheses When the null hypothesis is true, we should reject it only α (e.g., 0.05) of the time. If our alternative hypothesis is non-directional, then we want to be able to reject the null if a sample value is either very large or very small.
95 Non-directional Hypotheses When the null hypothesis is true, we should reject it only α (e.g., 0.05) of the time. If our alternative hypothesis is non-directional, then we want to be able to reject the null if a sample value is either very large or very small. This means that we need to split the 0.05 probability in half, between very high and very low values.
96 Non-directional Hypotheses When the null hypothesis is true, we should reject it only α (e.g., 0.05) of the time. If our alternative hypothesis is non-directional, then we want to be able to reject the null if a sample value is either very large or very small. This means that we need to split the 0.05 probability in half, between very high and very low values. As a result, we will only reject H 0 if the number of heads is in the top 2.5% of the distribution, or in the bottom 2.5%
97 Non-directional Hypotheses When the null hypothesis is true, we should reject it only α (e.g., 0.05) of the time. If our alternative hypothesis is non-directional, then we want to be able to reject the null if a sample value is either very large or very small. This means that we need to split the 0.05 probability in half, between very high and very low values. As a result, we will only reject H 0 if the number of heads is in the top 2.5% of the distribution, or in the bottom 2.5% Overall, we will make a Type I Error at most 5% of the time that the null hypothesis is correct.
98 Non-directional Hypotheses Here s both sides at once. Probability P( X <= 227 OR X >= 273 ) = Values
99 Outline Two-Tailed Tests
100 Hypothesis Testing Procedure 1. Translate research question into quantitative null and alternative hypotheses (H 0 and H 1 ) about the population/long run. 2. Identify a test statistic (what quantity (random variable) are we going to use to summarize the data?) 3. Establish a rejection criterion. (a) Find the distribution that the test statistic would have if H 0 (or the version of H 0 closest to H 1 ) is true. (b) Decide which data values would favor H 1 over H 0. If H 1 is unidirectional, then values on one extreme of the distribution are supportive. If H 1 is bidirectional, then values on either extreme of the distribution are supportive. (c) Set corresponding cutoff(s) so that the (conditional) probability of a Type I Error is low enough (usually 5% or less). 4. Collect the data, compute the test statistic, and make the decision.
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