Stat 360 Regression Analysis

STAT 405 - BIOSTATISTICS

Handout 9 – Number Needed to Treat (NNT), Number Needed to Harm (NNH) and Attributable Risk (AR)

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 100 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 20 out of 100 people.

Source: http://www.evidence-based-medicine.co.uk/ebmfiles/WhatisanNNT.pdf

Questions:

1. What is the ARR?

2. 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 10,000 men have no thrombolytic treatment after a heart attack, and 1,000 die within six weeks. In contrast, of 10,000 men given a thrombolytic agent, the number dying within six weeks is reduced to 800.

Source: http://www.evidence-based-medicine.co.uk/ebmfiles/WhatisanNNT.pdf

Questions:

1. What is the ARR?

2. 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.

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:

Questions:

1. What does an NNT of 1 imply?

2. 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.

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

ARR±zα2p11-p1n1 +p2(1-p2)n2

This is the same the confidence interval for (p1-p2) 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 100 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 20 out of 100 people.

§ Find a 95% confidence interval for ARR:

§ Find a 95% confidence interval for NNT:

Another Method for Finding a Confidence Interval for the 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:

e1 = 70; /* Number who benefit in experimental group */

n1 = 100; /* Sample size of experimental group */

e2 = 20; /* Number who benefit in control group */

n2 = 100; /* Sample size of control group */

The rest of the program should remain untouched!

Output:

General Comments on the NNT and the Confidence Interval:

1. The Wilson method has been shown to have a higher coverage rate than the asymptotic method.

2. 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.

Number Needed to Harm (NNH)

For adverse effects, we can calculate a number needed to harm (NNH).

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 1991 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 / Subjects / Death/cardiac arrest
placebo / 743 / 26
flecainide or encainide / 755 / 63

Source: http://smbrower.com/mediawiki/index.php/Understanding_basic_statistics#Number_needed_to_harm

Questions:

1. Find the NNH.

2. Interpret this quantity.

3. Find a 95% confidence interval for the NNH.

Attributable Risk (AR)

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.

Let

RR = be the relative risk for a disease given the person has the risk factor

d = probability of disease for persons without the risk factor

then

______= probability of disease for persons with the risk factor.

Now suppose that

p = probability that a person has the risk factor

then the overall probability of disease is

pD=

If all persons with the risk factor become risk factor free what proportion of cases or percent of cases would be eliminated?

This is called the attributable risk (AR) which is usually expressed a percent.

AR =

Confidence Interval for AR (Equation 13.12, pg.603)

100%ec11+ec1, 100%ec21+ec2

Here (c1,c2) is the interval obtained from

y±z1-α2RRRR-1ban1+dcn2

and

y=lnAR100-AR

Note: There are a 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!

Example of AR drug example:

An example of a drug that was removed from the market due to a low NNH is encainide. A 1991 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 / Subjects / Death/cardiac arrest
placebo / 743 / 26
flecainide or encainide / 755 / 63

If all patients taking flecainide or encainide were moved to placebo what is the associated attributable risk (AR) and the associated CI?

> AR(63,755,26,743,p=.05)

Estimated AR = 6.006

CI for AR = ( 2.804 , 12.4 )

> AR(63,755,26,743,p=.20)

Estimated AR = 20.36

CI for AR = ( 10.35 , 36.15 )

> AR(63,755,26,743,p=.50)

Estimated AR = 38.99

CI for AR = ( 22.39 , 58.6 )

Comparing Cardiac Arrest Deaths (Drug vs. Placebo)

Contingency Table

Count
Row % / N / Y
Drug / 755
92.30 / 63
7.70 / 818
Placebo / 743
96.62 / 26
3.38 / 769
1498 / 89 / 1587
Fisher's Exact Test / Prob / Alternative Hypothesis /
Left / 0.0001* / Prob(Death/Cardiac Arrest=Y) is greater for Group=Drug than Placebo
Right / 1.0000 / Prob(Death/Cardiac Arrest=Y) is greater for Group=Placebo than Drug
2-Tail / 0.0002* / Prob(Death/Cardiac Arrest=Y) is different across Group

Relative Risk

Description / Relative Risk / Lower 95% / Upper 95% /
P(Y|Drug)/P(Y|Placebo) / 2.277929 / 1.458103 / 3.558707

Odds Ratio

Odds Ratio / Lower 95% / Upper 95% /
0.419364 / 0.26264 / 0.669609

Two Sample Test for Proportions

In R we can write simple programs for inference for RR, OR, and AR

You can simply copy and paste this code in R and these functions will be available to you forever as long as your workspace.

Odds Ratio (OR)

OR = function (a, b, c, d, alpha = 0.05)

{

OR <- a * d/(b * c)

logOR <- log(OR)

SElogOR <- sqrt((1/a) + (1/b) + (1/c) + (1/d))

marerr <- qnorm(1 - (alpha/2)) * SElogOR

LCL <- exp(logOR - marerr)

UCL <- exp(logOR + marerr)

cat("\n")

cat(paste("Estimated OR =", OR))

cat("\n")

cat(paste("CI for OR = (", format(LCL, dig = 4), ",", format(UCL,

dig = 4), ")"))

cat("\n")

}

Relative Risk (RR)
RR = function (a, b, c, d, alpha = 0.05)

{

RR <- (a/(a + b))/(c/(c + d))

logRR <- log(RR)

SElogRR <- sqrt((b/(a * (a + b))) + (d/(c * (c + d))))

marerr <- qnorm(1 - (alpha/2)) * SElogRR

LCL <- exp(logRR - marerr)

UCL <- exp(logRR + marerr)

cat("\n")

cat(paste("Estimated RR =", RR))

cat("\n")

cat(paste("CI for RR = (", format(LCL, dig = 4), ",", format(UCL,

dig = 4), ")"))

cat("\n")

}

Attributable Risk (AR)

AR = function(a,b,c,d,p=.5,alpha=.05) {

RR <- (a/(a+b))/(c/(c+d))

logRR <- log(RR)

SElogRR <- sqrt((b/(a*(a+b))) + (d/(c*(c+d))))

za2 = qnorm(1 - (alpha/2))

AR = 100*(RR - 1)*p/((RR - 1)*p + 1)

y = log(AR/(100 - AR))

marerr <- za2*(RR/abs(RR-1))*SElogRR

c1 = y - marerr

c2 = y + marerr

LCL <- 100*exp(c1)/(1+exp(c1))

UCL <- 100*exp(c2)/(1+exp(c2))

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")

}