Unit 3: Foundations for inference Lecture 3: Decision errors, significance levels, sample size, power, and bootstrapping

Transcription

1 Unit 3: Foundations for inference Lecture 3: Decision errors, significance levels, sample size, power, and bootstrapping Statistics 101 Thomas Leininger June 3, 2013

2 Type 1 and Type 2 errors Decision errors Hypothesis tests are not flawless. In the court system innocent people are sometimes wrongly convicted and the guilty sometimes walk free. Similarly, we can make a wrong decision in statistical hypothesis tests as well. The difference is that we have the tools necessary to quantify how often we make errors in statistics. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

3 Type 1 and Type 2 errors Decision errors (cont.) There are two competing hypotheses: the null and the alternative. In a hypothesis test, we make a decision about which might be true, but our choice might be incorrect. Decision fail to reject H 0 reject H 0 H 0 true Type 1 Error Truth HA true Type 2 Error A Type 1 Error is rejecting the null hypothesis when H 0 is true. A Type 2 Error is failing to reject the null hypothesis when H A is true. We (almost) never know if H 0 or H A is true, but we need to consider all possibilities. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

4 Type 1 and Type 2 errors Hypothesis Test as a trial If we again think of a hypothesis test as a criminal trial then it makes sense to frame the verdict in terms of the null and alternative hypotheses: H 0 : Defendant is innocent H A : Defendant is guilty Which type of error is being committed in the following cirumstances? Declaring the defendant innocent when they are actually guilty Declaring the defendant guilty when they are actually innocent Which error do you think is the worse error to make? better that ten guilty persons escape than that one innocent suffer William Blackstone Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

5 Error rates & power Type 1 error rate As a general rule we reject H 0 when the p-value is less than 0.05, i.e. we use a significance level of 0.05, α = This means that, for those cases where H 0 is actually true, we do not want to incorrectly reject it more than 5% of those times. In other words, when using a 5% significance level there is about 5% chance of making a Type 1 error. P(Type 1 error) = α This is why we prefer to small values of α increasing α increases the Type 1 error rate. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

6 Error rates & power Filling in the table... Decision fail to reject H 0 reject H 0 H 0 true 1 α Type 1 Error, α Truth HA true Type 2 Error, β, 1 β Type 1 error is rejecting H 0 when you shouldn t have, and the probability of doing so is α (significance level) Type 2 error is failing to reject H 0 when you should have, and the probability of doing so is β (a little more complicated to calculate) of a test is the probability of correctly rejecting H 0, and the probability of doing so is 1 β In hypothesis testing, we want to keep α and β low, but there are inherent trade-offs. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

7 Error rates & power A quick example In a cancer screening, what happens if we conclude a patient has cancer and they do in fact have cancer? What if they didn t have cancer (but we concluded that they did)? What if we conclude the patient has cancer but we conclude that they do not have cancer? Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

8 Error rates & power Type 2 error rate If the alternative hypothesis is actually true, what is the chance that we make a Type 2 Error, i.e. we fail to reject the null hypothesis even when we should reject it? The answer is not obvious. If the true population average is very close to the null hypothesis value, it will be difficult to detect a difference (and reject H 0 ). If the true population average is very different from the null hypothesis value, it will be easier to detect a difference. Clearly, β depends on the effect size (δ) Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

9 Example - Blood Pressure Blood pressure oscillates with the beating of the heart, and the systolic pressure is defined as the peak pressure when a person is at rest. The average systolic blood pressure for people in the U.S. is about 130 mmhg with a standard deviation of about 25 mmhg. We are interested in finding out if the average blood pressure of employees at a certain company is greater than the national average, so we collect a random sample of 100 employees and measure their systolic blood pressure. What are the hypotheses? We ll start with a very specific question What is the power of this hypothesis test to correctly detect an increase of 2 mmhg in average blood pressure? Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

10 Problem 1 Which values of x represent sufficient evidence to reject H 0? (Remember H 0 : µ = 130, H A : µ > 130) Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

11 Problem 2 What is the probability that we would reject H 0 if x did come from N(mean = 132, SE = 2.5). Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

12 Putting it all together Null distribution Systolic blood pressure Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

13 Putting it all together Null distribution distribution Systolic blood pressure Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

14 Putting it all together Null distribution distribution Systolic blood pressure Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

15 Putting it all together Null distribution distribution Systolic blood pressure Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

16 Putting it all together Null distribution distribution Systolic blood pressure Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

17 Achieving desired power There are several ways to increase power (and hence decrease type 2 error rate): 1 Increase the sample size. 2 Decrease the standard deviation of the sample, which essentially has the same effect as increasing the sample size (it will decrease the standard error). 3 Increase α, which will make it more likely to reject H 0 (but note that this has the side effect of increasing the Type 1 error rate). 4 Consider a larger effect size δ. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

18 Choosing sample size for a particular margin of error If I want to predict the proportion of US voters who approve of President Obama and I want to have a margin of error of 2% or less, how many people do I need to sample? 1 Given desired error level m, we need m ME = z σ n. 2 To get m z σ n, I need n Note: This requires an estimate of σ. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

19 Bootstrapping Rent in Durham A random sample of 10 housing units were chosen on raleigh. craigslist.org after subsetting posts with the keyword durham. The dot plot below shows the distribution of the rents of these apartments. Can we apply the methods we have learned so far to construct a confidence interval using these data. Why or why not? rent Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

20 Bootstrapping Bootstrapping An alternative approach to constructing confidence intervals is bootstrapping. This term comes from the phrase pulling oneself up by one s bootstraps, which is a metaphor for accomplishing an impossible task without any outside help. In this case the impossible task is estimating a population parameter, and we ll accomplish it using data from only the given sample. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

21 Bootstrapping Bootstrapping Bootstrapping works as follows: (1) take a bootstrap sample - a random sample taken with replacement from the original sample, of the same size as the original sample (2) calculate the bootstrap statistic - a statistic such as mean, median, proportion, etc. computed on the bootstrap samples (3) repeat steps (1) and (2) many times to create a bootstrap distribution - a distribution of bootstrap statistics The 95% bootstrap confidence interval is estimated by the cutoff values for the middle 95% of the bootstrap distribution. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

22 Bootstrapping Rent in Durham - bootstrap interval The dot plot below shows the distribution of means of 100 bootstrap samples from the original sample. Estimate the 90% bootstrap confidence interval based on this bootstrap distribution bootstrap means Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

23 Randomization testing Randomization testing for a mean We can also use a simulation method to conduct the same test. This is very similar to bootstrapping, i.e. we randomly sample with replacement from the sample, but this time we shift the bootstrap distribution to be centered at the null value. The p-value is then defined as the proportion of simulations that yield a sample mean at least as favorable to the alternative hypothesis as the observed sample mean. Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

24 Randomization testing Rent in Durham - randomization testing According to rentjungle.com the average rent for an apartment in Durham is $854. Your random sample had a mean of $ Does this sample provide convincing evidence that the article s estimate is an underestimate? H 0 : µ = $854 H A : µ > $854 p-value: proportion of simulations where the simulated sample mean is at least as extreme as the one observed. 3 / 100 = randomization means Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

25 Randomization testing Extra Notes - Calculating Begin by picking a meaningful effect size δ and a significance level α Calculate the range of values for the point estimate beyond which you would reject H 0 at the chosen α level. Calculate the probability of observing a value from preceding step if the sample was derived from a population where x N(µ H0 + δ, SE) Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

26 Randomization testing Example - Using power to determine sample size Going back to the blood pressure example, how large a sample would you need if you wanted 90% power to detect a 4 mmhg increase in average blood pressure for the hypothesis that the population average is greater than 130 mmhg at α = 0.05? Given: H 0 : µ = 130, H A : µ > 130, α = 0.05, β = 0.10, σ = 25, δ = 4 Step 1: Determine the cutoff in order to reject H 0 at α = 0.05, we need a sample mean that will yield a Z score of at least x > n Step 2: Set the probability of obtaining the above x if the true population is centered at = 134 to the desired power, and solve for n. ( P x > ) = 0.9 n P Z > ( ) n 134 (Z = P > n ) n = Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23

27 Randomization testing Example - Using power to determine sample size (cont.) You can either directly solve for n, or use computation to calculate power for various n and determine the sample size that yields the desired power: power For n = 336, power = , therefore we need 336 subjects in our sample to achieve the desired level of power for the given circumstance. n Statistics 101 (Thomas Leininger) U3 - L4: Decision errors, significance levels, sample size, and power June 3, / 23