STAT 405 BIOSTATISTICS (Fall 2016) Handout 9 Number Needed to Treat, Number Needed to Harm, and Attributable Risk

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1 STAT 405 BIOSTATISTICS (Fall 06) The Absolute Risk Reduction (ARR) is the difference between the event rate in the experimental group and the event rate in the control group. In other words, this is the change in absolute risk brought about by an experimental intervention. The Number Needed to Treat (NNT) is the reciprocal of the ARR. EXAMPLE: An analgesic agent is given to 00 people, and 70 have their pain relieved within two hours. In contrast, the administration of a placebo tablet containing no active drug leads to pain relief in only 0 out of 00 people. Source: Questions:. What is the ARR?. What is the NNT? Interpretation: Two people must be given the analgesic for one of them to obtain effective pain relief. EXAMPLE: Consider the use of a thrombolytic agent after myocardial infarction. Say that 0,000 men have no thrombolytic treatment after a heart attack, and,000 die within six weeks. In contrast, of 0,000 men given a thrombolytic agent, the number dying within six weeks is reduced to 800. Source: Questions:. What is the ARR?. What is the NNT? Interpretation: Fifty people must be given the thrombolytic therapy after a heart attack to prevent one of them from dying within six weeks who would have died had they not been given thrombolysis.

2 STAT 405 BIOSTATISTICS (Fall 06) These interpretations lead to an alternative definition of NNT: The Number Needed to Treat (NNT) can also be defined as the number of persons who must be treated for a given period to achieve an event (treatment) or to prevent an event (prophylaxis). CALCULATING NNTs AND CONFIDENCE INTERVALS The NNT can be calculated from the simple formula: NNT = = ARR (Proportion who benefit from treatment - Proportion who benefit from control) NNT = = ARR (Proportion of adverse outcomes w/o trt - Proportion of adverse outcomes w/trt) Questions:. What does an NNT of imply?. If a treatment works well, do you expect to see a large NNT? Explain. We have already discussed the calculation of NNT for two examples. However, it should be noted that this point estimate is likely to change if we took another sample. Therefore, we should also report a confidence interval for NNT.

3 STAT 405 BIOSTATISTICS (Fall 06) The most commonly used method for constructing a confidence interval for NNT involves inverting and exchanging the confidence limits for the ARR. Confidence Interval for the ARR Let pˆ = the proportion who benefit in the treatment group and pˆ = the proportion who benefit in the control group (n and n are the group sample sizes, respectively). ARR ± z α pˆ ( - pˆ ) pˆ ( - pˆ ) + n n This is the same the confidence interval for (pp pp ) covered in introductory statistics courses. Confidence Interval for the NNT Simply invert and exchange the confidence limits for the ARR. EXAMPLE: Consider the example in which an analgesic agent is given to 00 people, and 70 have their pain relieved within two hours. In contrast, the administration of a placebo tablet containing no active drug leads to pain relief in only 0 out of 00 people. Find a 95% confidence interval for ARR: Find a 95% confidence interval for NNT: 3

4 STAT 405 BIOSTATISTICS (Fall 06) Using R to Find the Confidence Interval for the ARR and NNT You can use the prop.test function in R as follows. > num_relief <- c(70,0) > num_group <- c(00,00) > prop.test(num_relief,num_group,correct=false) Can omit this to employ the continuity correction -sample test for equality of proportions without continuity correction data: num_relief out of num_group X-squared = , df =, p-value =.89e- alternative hypothesis: two.sided 95 percent confidence interval: sample estimates: prop prop To find the enpdoints for a confidence interval for the NNT, you can invert and exchange the confidence limits for the ARR. Another Method for Finding a Confidence Interval for the ARR and NNT Ralf Bender, an epidemiologist, argues that the method used above to find a confidence interval for the ARR often leads to unreliable confidence limits. He argues that instead of using this method, one should use Wilson s score method for calculating the confidence interval for ARR. You can read more in his paper titled Improving the calculation of confidence intervals for the number needed to treat. He has made a SAS program available which calculates the ARR, NNT, and confidence intervals using both the unreliable method and his proposed method. The program is available in the file Bender_CIs.sas. To use this program, you need only enter your data as follows: e = 70; /* Number who benefit in experimental group */ n = 00; /* Sample size of experimental group */ e = 0; /* Number who benefit in control group */ n = 00; /* Sample size of control group */ The rest of the program should remain untouched! 4

5 STAT 405 BIOSTATISTICS (Fall 06) Output: General Comments on the NNT and the Confidence Interval:. The Wilson method has been shown to have a higher coverage rate than the asymptotic method.. The asymptotic method is most unreliable with very small sample sizes or with a very low absolute risk reduction (ARR). 3. Any estimated NNT should be accompanied by its confidence interval, and it is good practice to state which calculation method was used for the interval. 4. The only time it is appropriate to compare NNTs is if we have NNTs for different interventions for the same condition with the same outcome of interest. 5

6 STAT 405 BIOSTATISTICS (Fall 06) Number Needed to Harm (NNH) For adverse effects, we can calculate a number needed to harm (NNH). NNH = = ARR (Prop. adversely affected bytreatment - Prop. adversely affected by control) This can be regarded as the number of persons that must be given a treatment in order to cause harm to one patient that would not have otherwise been harmed. EXAMPLE: An example of a drug that was removed from the market due to a low NNH is encainide. A 99 study in the New England Journal of Medicine found proarrhythmic effects of encainide and flecainide were more likely in the antiarrhythmic group than in the placebo group. Data from the study are presented below: Treatment Total # of Subjects Death/cardiac arrest placebo flecainide or encainide Source: Questions:. Find the NNH.. Interpret this quantity. 3. Find a 95% confidence interval for the NNH. 6

7 STAT 405 BIOSTATISTICS (Fall 06) Attributable Risk (AR) From page 646 of your text: In some cases, a risk factor may have a large RR. However, if the risk factor is relatively rare, only a small proportion of cases may be attributable to this risk factor. Conversely, if a risk factor is common, then even a moderate RR may translate to a large number of cases attributable to the risk factor. The concept of attributable risk is useful in these circumstances. EXAMPLE: Cancer (Example 3.4 from your text) To see the intuition behind attributable risk, consider the following data table. A group of 0,000 current smoking women and 40,000 never-smoking women ages were followed for 5 years to see if they developed lung cancer. Lung Cancer No Lung Cancer Totals Smoker 50 9,950 0,000 Nonsmoker 0 39,990 40,000 Totals 60 49,940 50,000 One proposed formula for computing the relative risk is as follows: A AR =, where A + B + C A = Number of smokers who develop lung cancer, due to their smoking habit B = Number of smokers who develop lung cancer, but not due to their smoking habit C = Number of nonsmokers who develop lung cancer For these data, AR = This is called the attributable risk (AR), which is usually expressed a percent. We can interpret this quantity by saying that of lung cancer cases are attributable to smoking. 7

8 STAT 405 BIOSTATISTICS (Fall 06) We can also express the AR as follows: p(rr - ) AR = p(rr ) +, where p = the probability that a person has the risk factor RR = the relative risk for a disease for persons with the risk factor as compared to those without the risk factor Lung Cancer No Lung Cancer Totals Smoker 50 9,950 0,000 Nonsmoker 0 39,990 40,000 Totals 60 49,940 50,000 Using this formula, the AR could also be computed as follows: Confidence Interval for AR (Equation 3., pg. 648) 00% eecc eecc + ee cc, 00% + ee cc Here (cc, cc ) is the interval obtained from yy ± zz αα RRRR bb + dd, where yy = ln AAAA RRRR aann ccnn 00 AAAA Note: There are very nice derivations of the standard errors (variances) of the RR, OR, and AR used in the confidence intervals presented in the text. These derivations use the delta method which is covered in mathematical statistics (STAT 450/460). You should definitely examine the derivation of the var(rr), var(or), and var(ar) in the book if you have this background!. 8

9 STAT 405 BIOSTATISTICS (Fall 06) R function for inference on the AR The following R function can be used to compute the AR and its confidence interval. AR = function(a,b,c,d,alpha=.05) { RR <- (a/(a+b))/(c/(c+d)) p <- (a+b)/(a+b+c+d) logrr <- log(rr) SElogRR <- sqrt((b/(a*(a+b))) + (d/(c*(c+d)))) za = qnorm( - (alpha/)) AR = 00*(RR - )*p/((rr - )*p + ) y = log(ar/(00 - AR)) marerr <- za*(rr/abs(rr-))*selogrr c = y - marerr c = y + marerr LCL <- 00*exp(c)/(+exp(c)) UCL <- 00*exp(c)/(+exp(c)) cat("\n") cat(paste("estimated AR =",format(ar,dig=4))) cat("\n") cat(paste("ci for AR = (",format(lcl,dig=4),",",format(ucl,dig=4),")")) cat("\n\n") } Consider the data from the previous example. Lung Cancer No Lung Cancer Totals Smoker 50 9,950 0,000 Nonsmoker 0 39,990 40,000 Totals 60 49,940 50,000 Call the function as follows to find the attributable risk (AR) of lung cancer associated with smoking (and the CI). > AR(50,9950,0,39990) Estimated AR = 79.7 CI for AR = ( 65.04, ) 9