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Traps and pitfalls in medical statistics Arvid Sjölander

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26 april 2015Arvid Sjölander2 Motivating example You are involved in a project to find out if snus causes ulcer. A questionnaire is sent out to 300 randomly chosen subjects. 200 subjects respond: We can use the relative risk (RR) to measure the association between snus and ulcer: Can we safely conclude that snus prevents ulcer? Ulcer YesNoR Snus Yes228 2/30 0.07 No1715317/170=0.1

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26 april 2015Arvid Sjölander3 Outline Systematic errors Selection bias Confounding Randomization Reverse causation Random errors Confidence interval P-value Hypothesis test Significance level Power

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26 april 2015Arvid Sjölander4 One possible explanation It is a wide spread hypothesis that snus causes ulcer. Snus users who develop ulcer may therefore feel somewhat guilty, and may therefore be reluctant to participate in the study Hence, RR<1 may be (partly) explained by an underrepresentation of snus users with ulcer among the responders. This is a case of selection bias.

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26 april 2015Arvid Sjölander5 Selection bias We only observe the RR among the potential responders. The RR among the responders (observed) may not be equal to the population RR (unobserved). Population Potential non- responders Potential responders Sample

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26 april 2015Arvid Sjölander6 How do we avoid selection bias? Make sure that the sample is drawn randomly from the whole population of interest - must trace the non-responders. Send out the questionnaire again, follow up phone calls etc. Population Potential non- responders Potential responders Sample

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26 april 2015Arvid Sjölander7 Another possible explanation Because of age-trends, young people use snus more often than old people. For biological reasons, young people have a smaller risks for ulcer than old people. Hence, RR<1 may be (partly) explained by snus-users being in “better shape” than non-users. This is a case of confounding, and age is called a confounder.

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26 april 2015Arvid Sjölander8 Confounding The RR measures the association between snus and ulcer. The association depends on both the causal effect, and the influence of age. In particular, even in the absence of a causal effect, there will be an (inverse) association between snus and ulcer (RR 1). ?

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26 april 2015Arvid Sjölander9 How do we avoid confounding? At the design stage: randomization, i.e. assigning “snus” and “no snus” by “the flip of a coin”. + reliable; it eliminates the influence of all confounders. - expensive and possibly unethical. At the analysis stage: adjust (the observed association) for (the influence of) age, e.g. stratification, matching, regression modeling. + cheap and ethical. - not fully reliable; cannot adjust for unknown or unmeasured confounders. ?

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26 april 2015Arvid Sjölander10 Yet another explanation It is a wide spread hypothesis among physicians that snus causes and aggravates ulcer. Snus users who suffers from ulcer may therefore be advised by their physicians to quit. Hence, RR<1 may be (partly) explained by a tendency among people with ulcer to quit using snus. This is a case of reverse causation.

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26 april 2015Arvid Sjölander11 Reverse causation Reverse causation can be avoided by randomization. SnusUlcer ?

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26 april 2015Arvid Sjölander12 Systematic errors Selection bias, confounding, and reverse causation, are referred to as systematic errors, or bias. “You don’t measure what you are interested in”. How can you tell if your study is biased? You can’t! (At least not from the observed data). It is important to design the study carefully and “think ahead” to avoid bias. What may the reason be for potential response/non-response? How can we trace the non-responders? Which are the possible confounders? Do we need to randomize the study? Would randomization be ethical and practically possible?

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26 april 2015Arvid Sjölander13 Example cont’d Assume that we believe that the study is unbiased (no selection bias, no confounding and no reverse causation). Can we safely conclude that snus prevents ulcer? Ulcer YesNoR Snus Yes228 2/30 0.07 No1715317/170=0.1

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26 april 2015Arvid Sjölander14 Random errors True RR = observed RR? True RR observed RR! Population Sample True RRObserved RR=0.7

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26 april 2015Arvid Sjölander15 Confidence interval Where can we expect the true RR to be? The 95% Confidence Interval (CI) answers this question. It is a range of plausible values for the true RR. Example: RR=0.7, 95% CI: (0.5,0.9). The narrower CI, the less uncertainty in the true RR. The width of the CI depends on the sample size, the larger sample, the narrower CI. How do we compute a CI? Ask a statistician!

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CI for our data RR=0.7, 95% CI: (0.16,2.74). Conclusion? 26 april 2015Arvid Sjölander16 Ulcer YesNoR Snus Yes228 2/30 0.07 No1715317/170=0.1

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26 april 2015Arvid Sjölander17 P-value Often, we specifically want to know whether the true RR is equal to 1 (no association between snus and ulcer). The hypothesis that the true RR = 1 is called the “null hypothesis”; H 0. The p-value (p) is an objective measure of the strength of evidence in the observed data against H 0. 0 < p < 1. The smaller p-value, the stronger evidence against H 0. How do we compute p? Ask a statistician?

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Factors that determine the p-value What do you think p depends on? The sample size: the larger sample, the smaller p. The magnitude of the observed association: the stronger association, the smaller p. A common mistake: “The p-value is low, but the sample size is small so we cannot trust the results”. Yes you can! The p-value takes the sample size into account. Once the p-value is computed, the sample size carries no further information. 26 april 2015Arvid Sjölander18

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P-value for our data P = 0.81 Conclusion? 26 april 2015Arvid Sjölander19 Ulcer YesNoR Snus Yes228 2/30 0.07 No1715317/170=0.1

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Making a decision The p-value is an objective measure of the strenght of evidence against H 0. The smaller p-value, the stronger evidence against H 0. Sometimes, we have to make a formal decision of whether or not to reject H 0. This decision process is formally called hypothesis testing. We reject H 0 when the evidence against H 0 are “strong enough”. i.e. when the p-value is “small enough”. 26 april 2015Arvid Sjölander20

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Significance level The rejection threshold is called the significance level. E.g. “5% significance level” means that we have decided to reject H 0 if p<0.05. That we use a low significance level level means that we require strong evidence against H 0 for rejection. That we use a high significance level means that we are satisfied with weak evidence against H 0 for rejection. What is the advantage of using a low significance level? What about a high significance level? 26 april 2015Arvid Sjölander21

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A parallell to the court room H 0 = the prosecuted is innocent. p value = the strength of evidence against H 0. Low significance level = need strong evidence to condemn to jail. Few innocent in jail, but many guilty in freedom. High significance level = weak evidence sufficient to condemn to jail. Many guilty in jail, but many innocent in jail as well. 26 april 2015Arvid Sjölander22

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Type I and type II errors There is always a trade-off between the risk for type I and the risk for type II errors. Low significance level (difficult to reject H 0 ) small risk for type I errors, but large risk for type II errors. High significance level (easy to reject H 0 ) small risk for type II errors, but large risk for type I errors. By convention, we use 5% significance level (reject H 0 if p<0.05). 26 april 2015Arvid Sjölander23 H 0 is falseH 0 is true Reject H 0 OK Type I error (false positive) Don’t reject H 0 Type II error (false negative) OK

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Relation between significance level and type I errors In fact, the significance level = the risk for type I errors. If we follow the convention and use 5% significance level (reject H 0 if p<0.05) then we have 5% risk of type I errors. What does this mean, more concretely? 26 april 2015Arvid Sjölander24 H 0 is falseH 0 is true Reject H 0 OK Type I error (false positive) Don’t reject H 0 Type II error (false negative) OK Sig level

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Power Power = the chance of being able to reject H 0, when H 0 is false. Relation between significance level and power: High significance level (easy to reject H 0 ) high power. Low significance level (difficult to reject H 0 ) low power. 26 april 2015Arvid Sjölander25 H 0 is falseH 0 is true Reject H 0 OK Type I error (false positive) Don’t reject H 0 Type II error (false negative) OK Sig level Power

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Power calculations It is important to determine the power of the study before data is collected. That the power is low means that we will probably not find what we are looking for. Direct calculation of the power is beyond the scope of this course Ask a statistician! 26 april 2015Arvid Sjölander26

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Power calculations, cont’d Heuristically, the power of the study is determined by three factors: The significance level; higher significance level gives higher power. The true RR; stronger association gives higher power. The sample size; larger sample gives higher power. Typically, we want to have a power of at least 80%. In practice, the significance level is fixed at 5%. We also typically have an idea of what deviations from H 0 that are scientifically relevant to detect (e.g. RR > 1.5). We determine the sample size that we need, to have the desired power. 26 april 2015Arvid Sjölander27

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26 april 2015Arvid Sjölander28 Systematic vs random errors There are two qualitative differences between systematic and random errors. #1 Data can tell us if an observed association is possibly due to random errors - check the p-value. Data can never tell us if an observed association is due to systematic errors. #2 Uncertainty due to random errors can be reduced by increasing the sample size narrower confidence intervals. Systematic errors results from a poor study design, and can not be reduced by increasing the sample size.

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26 april 2015Arvid Sjölander29 Summary In medical research, we are often interested in the causal effect of one variable on another. An observed association between two variables does not necessarily imply that one causes the other. Always be aware of the following pitfalls: Selection bias Confounding Reverse causation Random errors

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