15 Risk and Odds
The analysis of the effects above depends mainly on the p-values and confidence
intervals of difference in proportions. There is a more common and often better
way of expressing these using Risk and Odds. In this chapter, we will use the
ANCData.txt data to illustrate these. First, we read the data
df_anc <-
read.delim("./Data/ANCData.txt") %>%
mutate(
death = factor(death, levels = c("no","yes"), labels = c("No", "Yes")),
clinic = factor(clinic),
anc = factor(anc, levels = c("old", "new"), labels = c("Old", "New")))And summarize the data
df_anc %>%
summarytools::dfSummary(graph.col = FALSE)
Data Frame Summary
df_anc
Dimensions: 755 x 3
Duplicates: 747
--------------------------------------------------------------------------
No Variable Stats / Values Freqs (% of Valid) Valid Missing
---- ---------- ---------------- -------------------- ---------- ---------
1 death 1. No 689 (91.3%) 755 0
[factor] 2. Yes 66 ( 8.7%) (100.0%) (0.0%)
2 anc 1. Old 419 (55.5%) 755 0
[factor] 2. New 336 (44.5%) (100.0%) (0.0%)
3 clinic 1. A 497 (65.8%) 755 0
[factor] 2. B 258 (34.2%) (100.0%) (0.0%)
--------------------------------------------------------------------------15.1 Risk
Risk is defined as the probability of having an outcome. Therefore, if in a the population of 100, 35 develop diabetes mellitus after a specified period of follow-up, the risk of developing diabetes in the population is
\[\frac{35}{100} = 0.35\] Tabulation of the ANC method and the occurrence of death below, we can conclude that the risk of perinatal mortality when one uses the old method is 0.11 (11.0%) and that for the new method is 0.06 (5.9%).
df_anc %>%
tbl_cross(percent = "col",
row = death,
col = anc,
label = list(death ~ "Death", anc = "ANC"),
digits = c(0,2)) %>%
bold_labels()| ANC | Total | ||
|---|---|---|---|
| Old | New | ||
| Death | |||
| No | 373 (89.02%) | 316 (94.05%) | 689 (91.26%) |
| Yes | 46 (10.98%) | 20 (5.95%) | 66 (8.74%) |
| Total | 419 (100.00%) | 336 (100.00%) | 755 (100.00%) |
This can be written as \[Re = 0:06 \text{ and } Rne = 0:11\]
Where \(Re\) is the risk in the exposed group (new anc method) and \(Rne\) is the risk in the non-exposed (old anc method).
15.2 Risk Ratio
A comparative way of expressing the risks in the two groups is by the use of the Risk Ratio or Relative Risk (RR). Where \[RR = \frac{Re}{Rne}\]
Note that by inference if the \(Re\) is the same as \(Rne\) then \(RR = 1\). The \(RR\) of perinatal mortality of the new compared to the old method is
\[RR = \frac{5.952381}{10.978520} = 0.5421843 \approx 0.54\]
The epiDisplay package has a function cs() which automatically calculates
the RR and other relevant stats with their confidence intervals. This is
applied to the ANCdata as below.
df_anc %$% epiDisplay::cs(outcome = death, exposure = anc, )
Exposure
Outcome Non-exposed Exposed Total
Negative 373 316 689
Positive 46 20 66
Total 419 336 755
Rne Re Rt
Risk 0.11 0.06 0.09
Estimate Lower95ci
Risk difference (Re - Rne) -0.05 -0.09
Risk ratio 0.54 0.32
Protective efficacy =(Rne-Re)/Rne*100 45.8 8.71
or percent of risk reduced
Number needed to treat (NNT) 19.9 10.79
or -1/(risk difference)
Upper95ci
-0.01
0.91
68.06
94.2
The output above first tabulates the two variables producing a contingency table with the marginal totals. It then shows our previously calculated parameters, Re and Rne. Rt (Risk total) is the risk if both the exposed and unexposed are put together, here.
\[Rt =\frac{20 + 46}{419 + 336} = \frac{66}{755} \approx 0.09\]
The next section of the output shows the risk difference (difference between the risks of the two groups), the risk ratio, the protective efficacy and the number needed to treat (NNT) together with their confidence intervals.
Interpreting the analysis so far, we conclude that the risk of perinatal death when using the new anc method is significantly less than using the old method. It significantly reduces the risk of death (Risk difference) by 0.05 (5%) and halves the chances of death (RR = 0.54, 95%CI: 0.32 to 0.91). About 20 (95%CI: 11 to 95) pregnant women need to be treated with the new anc method to prevent one perinatal death (NNT).
15.3 Odds
Another way of expressing the risk of an outcome is using the Odds. Statistically the odds is defined as
\[Odds = \frac{p}{1-p}\]
Where p is the probability of the outcome occurring. Using the ANCdata the probability of death in the exposed is.
\[pe = \frac{20}{336} = 0.05952381\]
The odds of death in the exposed can then be determined as
\[Oddse = \frac{0.05952381}{1-0.05952381} = 0.06329114\]
Similarly, the probability of death in the non-exposed (old anc type) is
\[pne = \frac{46}{419} = 0.1097852\]
And the odds of death in the non-exposed is
\[Oddsne = \frac{0.1097852}{1-0.1097852} = 0.1233244\]
15.4 Odds ratio
The comparative way of comparing the two odds is the Odds Ratio (OR). This is determined as
\[OR = \frac{Oddse}{Oddsne} = 0.5132086 \approx 0.51\]
Once again fortunately we do not have to go through this tedious procedure each
time we need to calculate the OR. The cc() function in the epiDisplay
package does this very well. Below we apply it to the analysis just done.
df_anc %$% epiDisplay::cc(outcome=death, exposure=anc, graph = FALSE)
anc
death Old New Total
No 373 316 689
Yes 46 20 66
Total 419 336 755
OR = 0.51
95% CI = 0.3, 0.89
Chi-squared = 5.9, 1 d.f., P value = 0.015
Fisher's exact test (2-sided) P value = 0.019 The output shows a table of the variables in question, the OR with its 95% confidence interval and both p-values determine by the chi-squared test and the Fisher’s test. With a confidence interval of the odds ratio not containing the null value 1, and small p-values from both methods it can be concluded that the odds of death in mothers who used the new ANC method is about half (0.5) of those who used the old method and the probability of obtaining an OR this values if the null was true, is low (p-value = 0.019). Therefore, the use of the new anc method is associated with significantly better perinatal outcomes compared to the old.
Odds ratios are very important in regression analysis and will be dealt with in more detail in subsequent chapters.