An Introduction to Tables

Confounding and Effect Modification

Interpretation and Choices

Population characteristics

• p = Probability of an event of interest• for example: Probability of successful post op• probability is thought to be ‘conditional’ on

factors of interest• for example: pre-op treatments (coded 0 and 1)

• Question: Does the probability of success depend on the choice of pre-op treatment?

But the patients receive a several different operations

• Surgery types are then coded 1 and 2• Question: Does our previous question depend on

the type of surgery?• i.e. Does the comparison between treatments (with

regard to the probability of success) depend on the type of surgery?

• This is addressing whether surgery type is an effect modifier

For example

• . cs suc tr,by(surg)

• surg | RR [95% Conf. Interval] M-H Weight• -----------------+-------------------------------------------------• 1 | 2 .8342841 4.79453 4.545455 • 2 | 1.9 1.759944 2.051202 45.45455 • -----------------+-------------------------------------------------• Crude | .3861386 .3348359 .4453018 • M-H combined | 1.909091 1.71543 2.124615• -------------------------------------------------------------------• Test of homogeneity (M-H) chi2(1) = 0.026 Pr>chi2 = 0.8724

Notice:

• The surgery group specific risk ratios are nearly equal. (2 is ‘close’ to 1.9)

• The test of the null hypothesis of no effect modification (in Stata, it is called the test for homogeneity) has a p-value of 0.8724

• So, on the basis of this test, there is no evidence that surgery type is an effect modifier (0.386 is not ‘close’ to

Since there is no evidence that surgery type is an effect modifier:

• We can assess whether surgery type is a confounder

• We compare the ‘crude’ estimate of the risk ratio with the ‘adjusted’ estimate of the risk ratio ( 0.386 is not ‘close’ to 1.909)

• So, there is ‘evidence’ that surgery type is a confounder.

Have a look at the surgery type specific tables

• . cs suc tr if surg==1,exact

• | tr |• | Exposed Unexposed | Total• -----------------+------------------------+----------• Cases | 100 5 | 105• Noncases | 900 95 | 995• -----------------+------------------------+----------• Total | 1000 100 | 1100• | |• Risk | .1 .05 | .0954545• | |• | Point estimate | [95% Conf. Interval]• |------------------------+----------------------• Risk difference | .05 | .0034122 .0965878 • Risk ratio | 2 | .8342841 4.79453 • Attr. frac. ex. | .5 | -.1986325 .791429 • Attr. frac. pop | .4761905 |• +-----------------------------------------------• 1-sided Fisher's exact P = 0.0667• 2-sided Fisher's exact P = 0.1503

…and• . cs suc tr if surg==2,exact

• | tr |• | Exposed Unexposed | Total• -----------------+------------------------+----------• Cases | 95 500 | 595• Noncases | 5 500 | 505• -----------------+------------------------+----------• Total | 100 1000 | 1100• | |• Risk | .95 .5 | .5409091• | |• | Point estimate | [95% Conf. Interval]• |------------------------+----------------------• Risk difference | .45 | .3972264 .5027736 • Risk ratio | 1.9 | 1.759944 2.051202 • Attr. frac. ex. | .4736842 | .4318 .512481 • Attr. frac. pop | .0756303 |• +-----------------------------------------------• 1-sided Fisher's exact P = 0.0000• 2-sided Fisher's exact P = 0.0000

The previous 2 displays highlight the importance of looking at the actual data

• Should the surgery specific risk ratios be offered? • Look at the p-values in each group• Look at the width of the confidence intervals in

each group.• Is a test for homogenity enough here?

Risk Ratios

• Usually: D - Disease E - Exposure• Risk ratio: RR = Pr(D|E) / Pr(D|not E)• Estimates of RR are usually written:

• Crude: Adjusted:

• Stratum specific:

• we can say that the risk of disease with exposure is estimated to be X times the risk of disease without exposure (with a p-value and/or a CI)

RR

cr

RR adj

RR

2RR

1RR

X,RR isIf

Risk Difference

• RD = Pr(D|E) - Pr(D|not E)

• Estimate is:

• Crude, Adjusted, Stratum-specific….

• then the risk of disease with exposure is estimated to be X higher than the risk of disease without exposure

RD

,XRD isIf

Odds

• Odds(E|D) = Pr(E|D)/Pr(not E|D)

• for example, if Pr(E|D) = 2/3, then odds(E|D) = 2

• notice: odds risk

• odds can be any positive number

• risk (is a probability) and must be between 0 and 1

Odds Ratios

• OR = odds(E|D)/odds(E|not D)• most useful in case-control studies• OR can be any positive number• log(OR) = logit can be any number (positive or

negative)• logits provide a ‘natural’ outcome for modelling• estimates written:• crude, adjusted, stratum-specific...

OR

Interpretation of Odds Ratios

• we can say that the odds of exposure with disease is estimated to be X times the odds of exposure without disease (with a p-value and/or a CI)

• But we can also say that the odds of disease with exposure is estimated to be X times the odds of disease without exposure (with a p-value and/or a CI)

• Odds ratio’s magic property!

X,OR isIf