Biostatistics for the Clinician

Lesson 3: Clinical Decision Making in a Multivariable Environment 3.1 - 2

Biostatistics for the Clinician

Table 13.1 contains 25 cases -- the mothers weight in kilograms and the infant's birthweight in grams. The question concerns whether you can find some relationship between the mother's weight and the infant's birth weight that allows you to identify likely candidates for fetal alcohol syndrome.. Why would such a relationship be important? How could such a relationship between the mother's weight and the infant's birth weight help you?

You want to check for malnutrition with regard to the fetus. You're looking at fetal alcohol syndrome. So what's the problem? Why can't you just say - well this kid's underweight and that kid's overweight or at weight, so this mother used alcohol and this mother didn't? What else do you need? Why do you have to fool around with some kind of linear relationship between the mother's weight and the infant's birth weight?

Notice the equation at the bottom of the table. X in the equation is the mother's weight, so if you know the mother's weight you can plug it in here and predict the baby's birthweight. So you take the mom's weight in kilograms, plug it in for X, multiply it by 30.7926, add 1501.40 to it, and that provides the gram estimate of the infant's weight. The expected weights based on those computations appear in the fourth column from the left.

Figure 13.6 above shows what the graph looks like when plotted. The points on the line are the estimates of the babies' weights, so if the mom weighs 70 kilograms, for example, the baby's predicted weight is about 3800 grams. Babies with large negative residuals, that is, whose weights are far below the predicted weights, are those who turn out to be associated with fetal alcohol syndrome. So what you've got is a very powerful technique to take humans which have a lot of variability and remove some of that variability statistically, so you can focus on the aspects of health that are the focal concern.

Lesson 3: Clinical Decision Making in a Multivariable Environment 3.1 - 6

Biostatistics for the Clinician

You can see above that you've got the two columns of data on the left side of the screen. The mother's weight is in the first column. The infant's weight is in the second. The last last seven cases of data run off the bottom and so are not visible.

Any of a variety of other computer programs could be used, SPSS, MINITAB, and WinStat are some programs designed especially for high level statistical analyses. These programs allow you to compute a regression analysis in a few point and click steps.

When you choose the "Data Analysis" option from the "Tools" menu, as shown above, the following screen showing a Data Analysis dialogue window appears.

At this point you only need to press the "OK" button. The analysis would run to completion and a screen like the following would then appear.

So, with a flick of a button you'd get all these numbers. You can't see it yet, but there's a scatterplot showing the regression line. The "a" and "b" values for the equation have been computed and you can see they are approximately equal to the one's you've already seen in Table 13.3. So, you've got the intercept value of approximately 1501.3 and the slope of 30.79 to multiply the mother's weight by in the y = ax + b equation. There are also some R values toward the upper left. They tell you how effective your fit is. In particular, the important value is the R² value. The R² will tell you what percent of the infant's weight variability is explained by the regression line. You can see from the R² above that about 27% of the variability in the infant's weight is predictable from the mother's weight, not great, but not too bad either.

If you then scrolled over to the right you'd then see the scatterplot shown below.

Let's say you want to find the males that have a serious problem with serum cholesterol. In this case you have some assumptions that their healthy cholesterol is related to their weight and their blood pressure. You want to look at the serum cholesterol of these patients with weight and blood pressure taken into account. So, you would need to do a multiple linear regression like that done with height and sex as predictors of weight. Using the table, your dependent variable would be serum cholesterol and your independent variables would be weight in kilograms and systolic blood pressure in mm of mercury.

You can see the equation you get, just below the table. The x_1j is the j'th person's weight. The x_2j is the j'th person's systolic blood pressure. So, what if you wanted to compute the expected or predicted serum cholesterol of the person with the systolic blood pressure of 120? You find 4.06 times 60.1 and subtract that from 18.25. Then you add 3.2 times 120. That would give you the predicted value. Then you could compare the predicted value with the actual. You'd compute the difference (called the residual) between the two. You could then see whether, based on these two things, the serum cholesterol is unusually high or low. You could do the same sort of thing with every patient.

Of course, you don't need to know how the equation was derived to use it. You simply need to know whether it is valid. All you need to do is plug in the values of the independent variables. You'll get predictions for the dependent variable.

Lesson 3: Clinical Decision Making in a Multivariable Environment 3.1 - 9

Biostatistics for the Clinician

You've got a log of a risk ratio on the left where the p represents the probability of giving birth to a child. On the right, you see the age of the mother as the coefficient of b₁. The equation, in this case doesn't really represent results of real research, but it does represent what might be computed from some real data. In the equation, frequency of intercourse is IC. The SC is the father's sperm count. The use of birth control pills is BCP, and the use of an IUD is represented in the last term of the equation. Now, you might say that's a complicated looking equation. But, don't worry about it! The statisticians handle the computations.

If you take the probability of the event happening (of a birth) divided by 1 minus that probability, you're just looking at the ratio of it happening to not happening. You then take the log of it. On the right side of the equation, it then turns out that these b's turn out to be risk ratios. Risk ratios are real nice to work with. For example, if one of these b values was a 3, that would cause 3 times more babies, because these b's are risk ratios.

The point is that you should avoid getting bogged down thinking about the equation. Don't worry about the notation. Let the statisticians handle that. You take the results which have been validated and use them in your practice and you work together with a biostatistician in applying them in your research.

Lesson 3: Clinical Decision Making in a Multivariable Environment 3.1 - 10

Biostatistics for the Clinician

When you have many independent variables that you're taking into account, you need to have many subjects. The recommended number is at least 10 subjects per independent variable. So, if you have five variables that you're using as predictors, make sure you have at least 50 subjects. Don't look at any studies that have small numbers of people that use multiple independent variables. The sampling error will be too large. You can't expect data from such small groups to be representative. This is a rule to remember. Use it to evaluate published medical research studies. Apply it when you're doing experiments that may have multiple variables.

It's a very good validation technique. You don't just use the predictors within the original group, but you go get a different group and see how accurate the predictions are with the new data. If the predictions hold up well, you've demonstrated how stable or reliable they are when you go from group to group. These are the kinds of cautions you need to employ when you have multiple variables as predictors in regression situtations.

Lesson 1: Summary Measures of Data 3.1 - 11