3 Binary Variable
Variabel yang nilainya cuma ada 2 (sukses/gagal, ya/tidak, sehat/tidak).
With binary outcomes we can estimate the proportion who have an outcome (disease)
i.e. we can estimate the probability or risk that an individual in a given population will have/develop this outcome
Examples:
– What proportion of Australian adults have ischaemic heart disease?
3.1 Estimasi Proporsi \(\pi\)
sample proportion (p) \[p=\frac{d}{n}\]
Contoh:
- Sample size (n): 11,247
- Number with diabetes (d): 844
- Proportion
Percentage People with diabetes = 7.5% (or 75 per 1000 people).
3.2 Membandingkan 2 Proporsi
With binary outcomes we can compare estimates of the proportion with disease in different populations
i.e. we can examine if there is an association between an exposure and disease by comparing the proportions with disease in the exposed and the unexposed groups.
Examples:
- Are men more likely than women to have diabetes?
3.2.1 Contingency Table
| Exposure | Outcome (Yes) | Outcome (No) | Total |
|---|---|---|---|
| Yes | \(d_1\) | \(h_1\) | \(n_1\) |
| No | \(d_2\) | \(h_2\) | \(n_2\) |
| Total | \(d\) | \(h\) | \(n\) |
3.2.2 Contoh Kasus
You survey 330 people at the festival.
- 250 ate ice cream and 80 didn’t
- Of those that ate ice cream, 170 developed gastro and 80 didn’t
- Of those that didn’t eat ice cream, 10 developed gastro and 70 didn’t
| Eat Ice Cream | Gastro (Yes) | Gastro (No) | Total |
|---|---|---|---|
| Yes | \(170\) | \(80\) | \(250\) |
| No | \(10\) | \(70\) | \(80\) |
| Total | \(180\) | \(150\) | \(330\) |
Overall risk of gastroenteritis \[180/330 = 0.545 (54.5\%)\] Risk of gastroenteritis in those who ate ice cream = \[170/250 = .68 (68\%)\] Risk of gastroenteritis in those who did not eat ice cream = \[10/80 = .125 (12.5\%)\]
3.2.3 Risk Difference (RD)
\[ \text{Risk Difference (RD) = Risk in exposed – Risk in unexposed}\]
3.2.4 Relative Risk (RR)
\[ \text{Relative Risk (RR)} = \frac{\text{Risk in exposed}}{\text{Risk in unexposed}}\]
These are known as measures of effect size.
\[RD = 68\% – 12.5\% = 55.5\%\] i.e. The absolute difference in risk of GI disease between those who did and did not eat the ice-cream was 55.5 percentage points.
\[RR = \frac{68\%}{12.5\%} = 5.4\] i.e. Those who ate ice cream were 5.4 times more likely to develop GI symptoms than those who did not eat the ice cream.
3.2.5 Odds dan Odds Ratio
The odds of an event are the probability that it does happen, divided by the probability that it does not happen.
We used odds when dealing with the rare outcome.
| Eat Ice Cream | Gastro (Yes) | Gastro (No) | Total | Odds of Gastro |
|---|---|---|---|---|
| Yes | \(170\) | \(80\) | \(250\) | \(170/80=2.125\) |
| No | \(10\) | \(70\) | \(80\) | \(10/70=0.143\) |
| Total | \(180\) | \(150\) | \(330\) |
Thus,
\[\text{Odds Ratio}=\frac{\text{Odds (Exposure=1)}}{\text{Odds (Exposure=0)}}=\frac{2.125}{0.143}=14.9\]
Interpretation: The odds of gastroenteritis was nearly 15 times higher in those that ate ice cream compared to those that didn’t
3.2.6 Odds Ratio vs Risk Ratio
Risk Ratios are usually easier to interpret and understand…
BUT Odds Ratios have some nice properties:
- we can use them for case-control studies (where there are usually not enough data to calculate risks)
- identical conclusions if we look at occurrence or absence of an outcome (e.g. odds of case versus odds of control)
- don’t run into computational difficulties with common outcomes
- for a rare outcome, the Odds Ratio is very close to the Risk Ratio