One of the main aims of statistical analysis is to be able to identify factors for the purpose of prediction, although it should be noted at the outset that prediction does not necessarily imply causation. In the context of adult anthropometry an obvious question to ask is 'How good a predictor is maternal BMI of pregnancy outcome (either child death or low birth weight)?'
Use of cut-off points
Frequently researchers decide to use a particular cut-off point. This can either be based on a number of normal deviates (Z scores) above and below the mean (usually either ±2 SD or +1.96 SD) or on the basis of some alternative criterion or criteria. In the context of BMI it is necessary to decide what constitutes 'low' BMI. Recently Ferro-Luzzi et al. (1992) used cut-offs of <16, 16.0-16.9, 17.0-18.4, 18.5-24.9, 25-29.9, 30.0-39.9 and 40+. These cut-offs were not arbitrary but judgmental (see Ferro-Luzzi et al. for details).
Table 7. Relative risk and odds ratio (OR) for the relationship between maternal BMI and the risk of a baby dying
No. of children | |||
Maternal BMI |
Died |
Survived |
Total |
<16 |
11 (a) |
51 (b) |
62 (a +b) |
(16 |
334 (c) |
3754 (d) |
4088 (c +d) |
Total |
345 (a +c) |
3805 (b +d) |
4150 (t) |
_{}
=_{}
_{}
Confidence limits for the OR:
1. Calculate loge OR = loge 2.43 = 0.888
2. Calculate the estimated standard error (SE):
_{}
3. The 95% confidence limits of OR:
_{}
log_{e} OR ± 1.96 = 0.888 ± 0.337
95% confidence limit on loge OR = 0.551 and 1.225
95% confidence limit on OR = e^{0.551} and e^{1.225 }= 1.73 and 3.40
Table 8. Sensitivity and specificity
No. of children | |||
Maternal BMI |
Died |
Survived |
Total |
<16 |
11 (a) |
51 (b) |
62 (a +b) |
> 16 |
334 (c) |
3754 (d) |
4088 (c +d) |
Total |
345 (a +c) |
3805 (b +d) |
4150 (t) |
True condition | |||
Test |
Died |
Survived |
Total |
< 16 |
True positive rate |
False positive rate |
Indicator positive rate |
2 16 |
False negative rate |
True negative rate |
Indicator negative |
Total |
True positive |
Total negative |
Total |
The true positive rate = ala + c = 11/354 = 3.2. This rate is also called the sensitivity of the test. The true negative rate = d/b + d = 3754/3805 = 98.7. This rate is also called the specificity of the test. Positive predictive value = the percentage of the indicator positives who are true positives = a/a + b = 17.7.
Use of c ^{2} test
Table 7 shows the number of child deaths in relation to a cut-off of <16 maternal BMI. This is the lowest cut-off of Ferro-Luzzi et al. and is very close to 1.96 SD below the mean for the Bangladeshi sample. The c ^{2} test (c ^{2} = 7.34, P < 0.01) shows significant discrepancy between observed and expected results with an excess of deaths in the <16 BMI category (observed 11, expected 5). Even so mothers with BMI of <16 only account for 11 out of 345 deaths.
Relative risk and odds ratio
To measure the strength of the association between the risk factor (<16 BMI) and the condition (death) the relative risk (RR) and/or the odds ratio (OR) can be calculated. Both ratios can take values between 0 and infinity. A value >1 signifies that the risk or odds of death are greater when exposed to the risk factor (positive association); a value of 0 indicates no association and a value of <1 indicates reduced risk or odds of death with exposure to the risk factor (negative association). Numerical differences in the RR and OR may occur but they always point in the same direction.
The RR for the data shown in Table 7 shows that mothers with BMI of <16 are 2.16 times as likely to have a child death than mothers with
BMI of 216. The OR has advantages over the relative risk since it is not dependent on how rare the condition is in the population and it provides consistent results whether a study is concerned with death or survival. The estimated OR for the Bangladeshi data is 2.43 (odds of a child death for a mother of <16 BMI compared to one ³ 16 BMI). The 95% confidence interval for or can also be calculated using Woolf's method and gives values between 1.73 and 3.40 (see Table 7).
Sensitivity and specificity
Another way of looking at the prediction is in terms of sensitivity and specificity. Screening tests use a procedure based on 'lot quantity assurance sampling'. As the name implies the method originated in sampling and inspecting manufactured goods when the purchaser does not want to accept a batch of goods with more than a certain percentage defective and the manufacturer does not want to reject the batch unless a certain percentage are defective. Table 8 uses the same Bangladeshi data as shown in Table 7 classified according to the screening test (BMI) and true condition (died/survived). A number of terms are calculated of which the sensitivity, specificity and positive predictive value are most commonly cited.
A good predictor is one which has a high sensitivity and high specificity. In addition a low false negative rate is important. Sensitivity and specificity are dependent on one another; high sensitivity is required with the identification of all the child deaths. Unfortunately this leads to lowered specificity and a high false positive rate and results in incorrectly identifying women as high risk. A high false positive rate is not as serious as a high false negative rate (i.e. failing to identify women as high risk) but it will burden the screening system.
The obvious limitations to the use of sensitivity and specificity are that the measures depend on the prevalences of the condition (child death) and on the risk factor (BMI). If the prevalence rates differ between studies then it is illegitimate to compare sensitivities. In addition the sensitivity and specificity measures are constrained by what cut-off value (s) have been chosen.
The example in Table 7 used 16 BMI as the cut-off. This gave a low sensitivity of 3.2, a high false negative rate and a high specificity. Clearly this cut-off value provides a poor prediction of child death.
An approach which gives equal weighting to the two forms of errors was developed by Youden (1950). He proposed an index J = 1 - (a + b) where a is the false negative rate and b is the false positive rate. If the test has no diagnostic value, a = 1 - b and J = 0. If the test is invariably correct, a = 1- b and J = 1 (values of J between -1 and 0 could arise if the test result were negatively associated with the true diagnosis, but this situation is unlikely to arise in practice). Thus the closer J is to one, the better
The data in Table 7 give a J value of 0.02. Higher values of J in the range 0.131 and 0.149 are found for BMIs between 20.5 and 21.5. Although Youden's index has been cited by some authors studying anthropometric measurements for predictive purposes, its use is limited since the two types of error are not treated equally - as noted above it is more important to maintain a low false negative rate at the expense of a higher false positive rate. Another method which is occasionally cited is based on the summation of the sensitivity and specificity rates; this suffers from the same disadvantage as Youden's index in that it gives equal weighting to both rates.
Shetty & James (1994) have suggested that a cut-off at 18.5 would be appropriate for both males and females. For the Bangladeshi data this would produce a sensitivity of 27, specificity of 86 and a positive predictive value of 15 for child death. For birth weight, defining low birth weight as <2500 g, the same BMI cut-off of 18.5 yields a sensitivity of 18, specificity of 96 and positive predictive value of 26. Thus this cut-off does not provide particularly good prediction since sensitivity is low and the number of false negatives is relatively high.
Table 9. Use of logistic regression analysis
Variable |
B |
Wald |
P |
BMI |
-0.1269 |
20.64 |
0.0001 |
MUAC |
-0.0553 |
2.98 |
0.0842 |
Age |
0.0763 |
51.52 |
0.0001 |
CPD |
0.6234 |
16.95 |
0.0001 |
Constant |
0.5103 |
0.164 |
0.425 |
^{c 2} = 105.034, d.f. = 4, P < 0 001
_{}
where z = B_{0} + B_{1}X_{1} + B_{2}X_{2}
z = -0.5103 + (-0.1269*BMI) + (0.0553*MUAC) + (0.0763*Age) + (0.6234*CPD)
For BMI = 17, MUAC = 20, Age = 16 and CPD = 1
(inadequate)
z = -2.2654; probability of death = 0.09
For BMI = 25, MUAC = 25, Age = 25 and CPD = 0
(adequate)
z = -3.1578, probability of death = 0.04
BMI = body mass index; MUAC = mid-upper arm circumference; CPD = cephalopelvic disproportion.
A better understanding of which cut-off to use can be obtained by plotting the sensitivity and specificity curves. Figure 1 presents these curves for BMIs and child death increasing at 0.5 intervals from 16 up to 25.5. Keeping in mind the aim (which is for an acceptable combination of higher sensitivity and low false negative rate) then BMIs in the range between 22 to 24 provide high specificity and lowish false negative rates. Of course the problem with such cut-offs in this range is that they result in a high false positive rate and the positive predictive value is only ±8.6%.
Logistic regression analysis
In the examples so far, one predictor variable, BMI, has been used as a screening tool. In practice researchers may want to evaluate the impact of a number of potential predictors in relation to child death, for example taking into account the effects of maternal age, BMI, midupper arm circumference (MUAC) and cephalopelvic disproportion (CPD, where the pelvis is too narrow to permit the passage of the baby and the obstructed labour can lead to death of the mother and/or child). The multivariate technique of logistic regression is designed to test for the relationship between a binary dependent variable (such as death or no death, or low birth weight and normal) and a number of independent variables which can be discrete (with many categories) and/or continuous. Thus the logistic regression analysis directly estimates the probability of an event occurring.
For more than one independent variable the model can be written as:
_{}
where z = B_{0} + B_{1}X_{1} + B_{2}X_{2}...
where e is the natural logarithm and B_{o} + B_{1} + B_{2} refer to the coefficients for each independent variable. Table 9 presents the results of a logistic regression analysis for birth outcome by maternal age, BMI, MUAC and CPD. It can be seen that even taking 'poor' values of these variables there is only a small chance of a child death; with higher values of BMI and MUAC and with older mothers with adequate CPD the risk becomes even lower.
Thus researchers should be encouraged to use logistic regression since it does not have the limitations of sensitivity and specificity measures. Furthermore it allows the researcher to test for the simultaneous effects of a number of discrete or continuous variables.
Discriminant analysis
The extent to which it would be possible to predict whether a child will die or survive depends on the degree of overlap between the two sample distributions. As the difference between the two sample means increases so does the separation of the samples. Figure 1 presents the generalized case of a variable, X, in which differences in means exist between two populations with the same variance.a is the false negative rate and b is the false positive rate. For any given a the value of b depends solely on the standardized distance, Q between the means. The equation is:
_{}
Table 10. Use of standardized distances (Q)
_{} | |||
• Weight | |||
Birth outcome |
n |
Mean |
SD |
Child died |
345 |
45.33 |
6.71 |
Child survived |
3805 |
48.40 |
6.92 |
Total |
4105 |
48.14 |
6.96 |
• Height | |||
Birth outcome |
n |
Mean |
SD |
Child died |
345 |
149.10 |
5.25 |
Child survived |
3805 |
150.83 |
5.11 |
Total |
4105 |
150.68 |
5.14 |
• Mid-upper arm circumference (MUAC) | |||
Birth outcome |
n |
Mean |
SD |
Child died |
345 |
22.56 |
2.33 |
Child survived |
3805 |
23.21 |
2.39 |
Total |
4105 |
23.15 |
2.39 |
Standardized distances | |||
Variable |
Q | ||
Weight |
0.45 | ||
BMI |
0.33 | ||
Height |
0.33 | ||
MUAC |
0.28 |
Table 11. Discriminant analysis pregnancy outcome by education, gravidity, mother's age and BMI
Predicted | ||
No death |
Death | |
Actual | ||
No death |
67.7 |
32.3 |
Death |
38.6 |
61.4 |
Overall correctly classified 67.2%
A critical value for the test is the mid-point between the means [e.g ½ (D _{1}-D _{2})] The merits of different tests can be determined by comparing their values of Q. Tests with high values of Q will differentiate between the two populations better than low values. Table 10 presents an example of the use of this standardized distance using MUAC, weight, height and BMI for birth outcome (death/no death). For child death, weight is the best predictor as the difference between means is close to half a standardized distance. For BMI and height the difference is one-third and for MUAC just over one-quarter of a standardized distance.
The standardized distances are based on the use of a single continuous variable as a predictor. When a number of continuous variables are to be used in combination then the discriminant function should be calculated. Most statistical packages have discriminant function routines and there is no need to calculate the function by hand. The principle of the method involves calculating the within-groups covariance matrix and its inverse together with the mean differences for each of the two variables. From these values the discriminant function can be calculated (a simple worked example is shown in Armitage & Berry (1987, pp. 338-9) for two continuous variables).
Table 12. Use of multiple regression analysis
1. Birth weight by BMI | |||||
Item |
d.f. |
Sum of square |
Mean squares | ||
Regression |
1 |
34717365.20 |
34717365.20 | ||
Residual |
4148 |
856366809.62 |
206452.94 | ||
F = 168.16, P < 0.0001 | |||||
Variable |
B |
SE B |
Beta |
t |
P (t) |
BMI |
33.93 |
2.62 |
0.197 |
12.97 |
0.0000 |
(Constant) |
2119.80 |
55.86 |
37.95 |
0.0000 |
2. Test of curvilinearity | |||
Step 2. Enter BMI^{2} | |||
d.f. |
Sum of square |
Mean. squares | |
Regression |
2 |
34734812.48 |
17367406.24 |
Residual |
4147 |
856349362.34 |
206498.52 |
F = 84.10426, P < 0.0001 |
The effect of BMI^{2} is assessed by subtracting the sum of squares at Step 2 from Step I = 17447.27 and by dividing this total by the residual mean square at Step 2.
F = 17447.27/206498.52 (not significant)
Therefore there is no significant curvilinearity since BMI^{2} is not significant.
3. Multiple regression analysis
Step 1: MUAC and foot length | |||||
d.f. |
Sum of squares |
Mean square | |||
Regression |
3 |
41262859.89 |
13754286.63 | ||
Residual |
4146 |
849821314.93 |
204973.79 | ||
F = 67.10, P < 0.0001 | |||||
Variable |
B |
SE B |
Beta |
t |
P |
BMI |
25.24 |
3.21 |
0.147 |
7.860 |
0.0001 |
MUAC |
15.10 |
3.77 |
0.078 |
4.004 |
0.0001 |
Foot |
17.95 |
6.75 |
0.042 |
2.657 |
0.0079 |
(Constant) |
1559.22 |
146.51 |
10.643 |
0.0000 |
Using the Bangladeshi dataset BMI, together with age of the mother, gravidity and educational status showed that over 67% of the child deaths could be predicted correctly (Table 11).
Modelling using multiple regression analysis
Finally there are times when it is of interest to treat birth weight as a continuous variable and examine its relationship to BMI. A simple regression analysis shows that there is a highly significant positive relationship in the Bangladeshi data set between birth weight and BMI and that for each increase in 1 unit in BMI, birth weight increases by, on average, nearly 34g (Table 12). However, even though the relationship is significant there is considerable variation around the regression line as can be seen in Fig. 2 and from the coefficient of determination R2 which was 0.039, i.e. BMI explains only 3.9% of the variance of birth weight.
A reasonable question to ask is whether there is any curvilinear relationship between birth weight and BMI. In the simple regression the equation was Y = a + bX, where a is the intercept value and b the regression coefficient. For test of curvilinearity the equation Y = a + bX + bX^{2}... is used, where bX^{2} refers to the quadratic term. Higher order terms, cubic, quartic etc., could be fitted. In the Bangladeshi data set there was no evidence of a significant curvilinear effect.
The modelling can become more complex and multiple regression analysis can be used to examine how much variation can be explained using a large number of continuous and discrete variables. If discrete variables are used with two categories they must be coded as O and 1. If discrete variables with more than two categories are to be used, dummy variables must be created with binary coding of 0,1. The number of dummy variables corresponds to n1 categories.
To illustrate the use of multiple regression an analysis of birth weight (the continuous dependent variable) was examined in relation to BMI, MUAC and foot length. The analysis showed that all three independent variables related significantly to birth weight (Table 12). Even so, R2 only increased to 4.7% from the value of 3.9% when BMI was the only independent variable.