# Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

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1 Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters without measuring the entire population. We can, however, make inferences about the population parameters from random samples. Last week, we saw how we can create a confidence interval, within which we are reasonably certain the population parameter lies. This week, we will see a different type of inference: can we be certain that the parameter does not take a particular value? Hypothesis Testing Significance Testing Form the Null Hypothesis Calculate probability of observing data if null hypothesis is true (p-value) Low p-value taken as evidence that null hypothesis is unlikely Originally, only intended as informal guide to strength of evidence against null hypothesis Fisher s p-value was very informal way to assess evidence against null hypothesis Neyman and Pearson developed more formal approach: significance testing Based on decision making: rule for deciding whether or not to reject the null hypothesis Clinically, need to make decisions. Scientifically, may be more appropriate to retain uncertainty. Introduces concepts of power, significance

2 The Null Hypothesis The Alternative Hypothesis Simplest acceptable model. If the null hypothesis is true, the world is uninteresting. Must be possible to epress numerically ( test statistic ). Sampling distribution of test statistic must be known. Null Hypothesis is untrue Covers any other possibility. May be one-sided, if effect in opposite direction is as uninteresting as the null hypothesis One and Two-sided tests Test Statistic Good eample: χ 2 test. χ 2 test measures difference between epected and observed frequencies Only unusually large differences are evidence against null hypothesis. Bad eample: clinical trial A drug company may only be interested in how much better its drug is than the competition. Easier to get a significant difference with a one-sided test. The rest of the world is interested in differences in either direction, want to see a two-sided test. One-sided tests are rarely justified Null hypothesis distribution must be known. Epected value if null hypothesis is true. Variation due to sampling error (standard error) if null hypothesis is true. From this distribution, probability of any given value can be calculated. Can be a mean, proportion, correlation coefficient, regression coefficient etc.

3 Normally Distributed Statistics Test statistic: Normal distribution Many test statistics can be considered normally distributed, if sample is large enough. If the test statistic T has mean µ and standard error σ, then T µ σ has a normal distribution with mean 0 and standard error 1. We do not know σ, we only have estimate s. If our sample is of size n, T µ has a t-distribution with s n 1 d.f. Hence the term t-test. If n 100, a normal distribution is indistinguishable from the t-distribution. Etreme values less unlikely with a t-distribution than a normal distribution. y T-distribution and Normal Distribution Non-Normally Distributed Statistics y Statistics may follow a distribution other than the normal distribution. χ 2 Mann-Whitney U Many will be normally distributed in large enough samples Tables can be used for small samples. Can be compared to quantiles of their own distribution Normal t distribution (10 d.f.)

4 Test Statistic: χ 2 4 Eample 1: Height and Gender y Null hypothesis On average, men and women are the same height Alternative Hypothesis One gender tends to be taller than the other. Test Statistic Difference in mean height between men and women. One-Sided Hypotheses Men are taller than women Women are taller than men Eample 2: Drinks preferences The p-value Null hypothesis Equal numbers of people prefer Coke and Pepsi Alternative Hypothesis Most people prefer one drink to the other Test Statistic Several possibilities: Difference in proportions preferring each drink Ratio of proportions preferring each drink One-Sided Hypotheses More people prefer Coke More people prefer Pepsi Probability of obtaining a value of the test statistic at least as etreme as that observed, if the null hypothesis is true. Small value data obtained was unlikely to have occurred under null hypothesis Therefore, null hypothesis is probably not true. Originally intended as informal way to measure strength of evidence against null hypothesis It it not the probability that the null hypothesis is true.

5 Interpreting the p-value Factors Influencing p-value 0 p 1 Large p ( 0.2, say) no evidence against null hypothesis p 0.05 there is some evidence against null hypothesis Effect is statistically significant at the 5% level 0.05 is an arbitrary value: is very little different from Smaller p stronger evidence Effect Size: a big difference is easier to find than a small difference. Sample Size: The more subjects, the easier to find a difference Always report actual, not p < 0.05 or p > 0.05 NS is unforgivable No significant difference can mean no difference in population or Sample size was too small to be certain Statistically significant difference may not be clinically significant. Interpreting a significance test Meta-Analysis Eample Significance test asks Is the null hypothesis true Answers either Probably not or No comment Answers are interpreted as No or Yes Misinterpretation leads to incorrect conclusions Ten studies 50 uneposed and 50 eposed in each Prevalence 10% in uneposed, 15% in eposed True RR = 1.5

6 Meta-Analysis Results Getting it Wrong Study RR p-value Pooled Data There are two ways to get it wrong: The null hypothesis is true, we conclude that it isn t (Type I error). The null hypothesis is not true, we conclude that it is (Type II error). Type I Error (α) Type II Error (β) Null hypothesis is true, but there is evidence against it. 1 time in 20 that the null hypothesis is true, a statistically significant result at the 5% level will be obtained. The smaller the p-value, the less likely we are making a type I error. Testing several hypotheses at once increases the probability that at least one of them will be incorrectly found to be statistically significant. Several corrections are available for Multiple Testing, Bonferroni s is the most commonly used, easiest and least accurate. Null hypothesis is not true, but no evidence against it in our sample. Depends on study size: small studies less likely to detect an effect than large ones Depends on effect size: large effects are easier to detect than small ones Power of a study = 1 - β = Probability of detecting a given effect, if it eists. Some debate about whether correction for multiple testing

7 Testing Testing : Eample Can compare to a hypothetical value (e.g. 0). Sometimes called. Test statistic T = µ S.E.(). Compare T to a t-distribution on n 1 d.f. The following data are uterine weights (in mg) for a sample of 20 rats. Previous work suggests that the mean uterine weight for the stock from which the sample was drawn was 24mg. Does this sample confirm that suggestion? 9, 14, 15, 15, 16, 18, 18, 19, 19, 20, 21, 22, 22, 24, 24, 26, 27, 29, 30, 32 = 21.0 S.D.() = Testing : Solution One-Sample t-test in Stata S.E.() = = T = 24.0 S.E.() = = 2.27 Comparing to a t-distribution on 19 degrees of freedom gives a p-value of I.e if the stock had a mean uterine weight of 24mg, and we took repeated random samples, less than 4 times in 100 would a sample have such a low mean weight.. ttest = 24 One-sample t test Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] Degrees of freedom: 19 Ho: mean() = 24 Ha: mean < 24 Ha: mean!= 24 Ha: mean > 24 t = t = t = P < t = P > t = P > t =

8 The Unpaired (two-sample) T-Test Two-Sample t-test in Stata For comparing two means If we are comparing in a group of size n and y in a group of size n y, Null hypothesis is = ȳ Alternative hypothesis is ȳ Test statistic ȳ T = S.E. of (ȳ ) T is compared to a t distribution on n + n y 2 degrees of freedom You may need to test (sdtest) whether the standard deviation is the same in the two groups.. ttest nurseht, by(se) Two-sample t test with equal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] female male combined diff Degrees of freedom: 400 Ho: mean(female) - mean(male) = diff = 0 Ha: diff < 0 Ha: diff!= 0 Ha: diff > 0 t = t = t = P < t = P > t = P > t = If not, use the option unequal. Comparing Proportions Comparing Proportions in Stata. cs back_p se We wish to compare p 1 = a n 1, p 2 = b n 2 Null hypothesis: π 1 = π 2 = π Standard error of p 1 p 2 = π(1 π)( 1 n n 2 ) Estimate π by p = a+b n 1 +n 2 p 1 p 2 can be compared to a standard normal p(1 p)( ) n 1 n 2 distribution se Eposed Uneposed Total Cases Noncases Total Risk Point estimate [95% Conf. Interval] Risk difference Risk ratio Attr. frac. e Attr. frac. pop chi2(1) = Pr>chi2 =

9 y Sample Size in stata Power Calculations Illustrated: 1 in stata Given: Null hypothesis value Alternative hypothesis value Standard error Significance level (generally 5%) Calculate: Power to reject null hypothesis for given sample size Sample size to give chosen power to reject null hypothesis H0 Shaded area = Sample value significantly different from H 0 = probability of type I error (If H 0 is true) Power Calculations Illustrated: 2 in stata y Power Calculations Illustrated: 3 in stata y H0 Ha H0 Ha H 0 : = 0, S.E.() = 1 H A : = 3, S.E.() = 1 Power = 85% H 0 : = 0, S.E.() = 1 H A : = 4, S.E.() = 1 Power = 98%

10 y Power Calculations Illustrated: 4 in stata Power Calculations in Stata in stata H0 Ha sampsi Only for differences between two groups Difference in proportion or mean of normally distributed variable Can calculate sample size for given power, or power for given sample size Does not account for matching H 0 : = 0, S.E.() = 0.77 Need hypothesised proportion or mean & SD in each group H A : = 3, S.E.() = 0.77 Power = 97% in stata Parameters in Sample Size Calculation sampsi synta for sample size in stata Power Significance level Mean (proportion) in each group SD in each group Missing SD proportion sampsi #1 #2, [ratio() sd1() sd2() power()] where ratio Ratio of the size of group 2 to size of group 1 (defaults to 1). sd1 Standard deviation in group 1 (not given for proportions). sd2 Standard deviation in group 2 (not given for proportions). Assumed equal to sd1 if not given). power Desired power as a probability (i.e. 80% power = 0.8). Default is 90%.

11 in stata in stata sampsi for power sampsi eamples sampsi #1 #2, [ratio() sd1() sd2() n1() n2()] where ratio Ratio of the size of group 2 to size of group 1 (defaults to 1). sd1 Standard deviation in group 1 (not given for proportions). sd2 Standard deviation in group 2 (not given for proportions). Assumed equal to sd1 if not given). n1 Size of group 1 n2 Size of group 2 Sample size needed to detect a difference between 25% prevalence in uneposed and 50% prevalence in eposed: sampsi Sample size needed to detect a difference in mean of 100 in group one and 120 in group 2 if the standard deviation is 20 and group 2 is twice as big as group 1 sampsi , sd1(20) ratio(2) in stata Criticism of hypothesis and significance testing Hypothesis and significance testing are complicated, convoluted and poorly understood Tell us a fact that is unlikely to be true about our population p-value depends on both effect size and study size: contains no eplicit information about either Intended as automated decision-making process: cannot include other information to inform decision Would prefer to know things that are true about the population More useful to have a range within which you believe a population parameter lies. Hypothesis Tests and Confidence Intervals Hypothesis tests about means and proportions are closely related to the corresponding confidence intervals for the mean and proportion. Null value outside 95% confidence interval null hypothesis rejected at 5% significance level. Confidence intervals convey more information and are to be preferred. Dichotomising at p = 0.05 ignores lots of information. p-value mies together information about sample size and effect size There is a movement to remove archaic hypothesis tests from epidemiology literature, to be replaced by confidence intervals. If there are several groups being compared, a single hypothesis test is possible, several confidence intervals

12 Hypothesis Tests vs. Confidence Intervals Outcome Eposed No Yes No 6 6 Yes 2 5 p = 0.37 OR = 2.5, 95% CI = (0.3, 18.3) Outcome Eposed No Yes No Yes p = 0.36 OR = 1.1, 95% CI = (0.9, 1.4)

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