Biomedical research often seeks to establish if there is a relationship between two variables; for example, is there a relationship between salt intake and blood pressure, or between cigarette smoking and life expectancy? The methods used to do this are correlational techniques, which focus on the "co-relatedness" of the two variables. There arc two basic kinds of correlational techniques:
• Correlation, which is used to establish and quantify the strength and direction of the relationship between two variables.
• Regression, which is used to express the functional relationship between two variables, so that the value of one variable can be predicted from knowledge of the other.
Correlation simply expresses the strength and direction of the relationship between two variables in terms of a correlation coefficient, signified by r. Values of r vary from —1 to +l;the strength of the relationship is indicated by the size of the coefficient, while its direction is indicated by the sign.
A plus sign means that there is a positive correlation between the two variables—high values of one variable (such as salt intake) are associated with high values of the other variable (such as blood pressure). A minus sign means that there is a negative correlation between the two variables—high values of one variable (such as cigarette consumption) are associated with low values of the other (such as life expectancy).
If there is a "pcrfect" linear relationship between the two variables, so that it is possible to know the exact value of one variable from knowledge of the other variable, the correlation coefficient (r) will be exactly plus or minus 1.00. If there is absolutely no relationship between the two variables, so that it is impossible to know anything about one variable on the basis of knowledge of the other variable, then the coefficient will be zero. Coefficients beyond ±0.5 are typically regarded as strong, whereas coefficients between zero and ±0.5 are usually regarded as weak.
The relationship between two correlated variables forms a bivariate distribution, which is commonly presented graphically in the form of a scattergram. The first variable (salt intake, cigarette consumption) is usually plotted on the horizontal (X) axis, and the second variable (blood pressure, life expectancy) is plotted on the vertical (Y) axis. Each plotted data point represents one observation of a pair of values, such as one patient's salt intake and blood pressure, so the number of plotted points is equal to the sample size n. Figure 4-1 shows four different scattergrams.
Determining a correlation coefficient involves mathematically finding the "line of best fit" to the plotted data points. The relationship between the appearance of the scattergram and the correlation
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coefficient can therefore be understood by imagining how well a straight line could fit the plotted points. In Figure 4-1 A, for example, it is not possible to draw any straight line that would fit the plotted points at all; therefore, the correlation coefficient is approximately zero. In Figure 4-1B, a straight line would fit the plotted points perfectly—so the correlation coefficient is 1.00. Figure 4-1C shows a strong negative correlation, with a correlation coefficient in the region of —0.8, and Figure 4-1D shows a weak positive correlation in the region of +0.3.
The two most commonly used correlation coefficients are the Pearson product-moment correlation, which is used for interval or ratio scale data, and the Spearman rank-order correlation, which is used for ordinal scale data. The latter is sometimes symbolized by the letter p (rho). Pearson's r would therefore be used (for example) to express the association between salt intake and blood pressure (which are both ratio scale data), whereas Spearman's p would be used to express the association between birth order and class position at school (which are both ordinal scale data).
Both these correlational techniques are linear: they evaluate the strength of a "straight line" relationship between the two variables. So if there is a very strong nonlinear relationship between two variables, the Pearson or Spearman correlation coefficients will be an underestimate of the true strength of the relationship.
Dose of drug
weak effects at very high or very low doses. Because the relationship between dose and effect is so nonlinear, the Pearson r correlation coefficient is low, even though there is actually a very strong relationship between the two variables. Visual inspection of scattergrams is therefore invaluable in identifying relationships of rhis sort. More advanced nonlinear correlational techniques can be used to quantify correlations of this kind.
Once a correlation coefficient has been determined, the coefficient of determination can be found by squaring the value of r. The coefficient of determination, symbolized by r2, expresses the proportion of the variance in one variable that is accounted for, or "explained," by the variance in the other variable. So if a study finds a correlation (r) of 0.40 between salt intake and blood pressure, it could be concluded that 0.40 X0.40 = 0.16, or 16% of the variance in blood pressure in this study is accounted for by variance in salt intake. This does not necessarily mean that the changes in salt intake caused the change in blood pressure.
A correlation between two variables does not demonstrate a causal relationship between the two variables, no matter how strong it is. Correlation is merely a measure of the variables' statistical association, not of their causal relationship. Inferring a causal relationship between two variables on the basis of a correlation is a common and fundamental error.
Furthermore, the fact that a correlation is present between two variables in a sample does not necessarily mean that the correlation actually exists in the population. When a correlation has been found between two variables in a sample, the researcher will normally wish to test the null hypothesis that there is no correlation between the two variables (i.e., that r = 0) in the population. This is done with a special form of t-test.
If two variables are highly correlated, it then becomes possible to predict the value of one of them (the dependent variable) from the value of other (the independent variable) by using regression techniques. In simple linear regression the value of one variable (X) is used to predict the value of the other variable (Y) by means of a simple linear mathematical function, the regression equation, which quantifies the straight-line relationship between the two variables. This straight line, or regression line, is actually the same "line of best fit" to the scattergram as that used in calculating the correlation coefficient.
The simple linear regression equation is the same as the equation for any straight line:
where a is a constant, known as the "intercept constant" because it is the point on the Y
axis where the Y axis is intercepted by the regression line;
b is the slope of the regression line, and is known as the regression coefficient; and
Once the values of a and b have been established, the expected value of Yean be predicted for any given value of X. For example, Zito and Reid (1978) showed that the hepatic clearance rate of lido-caine (Y, in ml/min/kg) can be predicted from the hepatic clearance rate of indocyanine green dye (X, in ml/min/kg), according to the equation Y = 0.30 + 1.07X, thus enabling anesthesiologists to reduce the risk of lidocaine overdosage by testing clearance of the dye.
Other techniques generate multiple regression equations, in which more than one variable is used to predict the expected value of Y, thus increasing the overall percentage of variance in Y that can be accounted for. For example, Rubin et al. (1986) found that the birth weight of a baby (Y, in grams) can be partly predicted from the number of cigarettes smoked on a daily basis by both the baby's mother (X,) and the baby's father (X2) according to the multiple regression equation Y = 3385 — 9X| — 6X2 Other techniques are available to quantify nonlinear relationships among multiple variables. As with correlation, however, it is important to remember that the existence of this kind of statistical association is not in itself evidence of causality.
CHOOSING AN APPROPRIATE INFERENTIAL OR CORRELATIONAL TECHNIQUE
The basic choice of an appropriate statistical technique for a particular research problem is determined by two factors: the scale of measurement and the type of question being asked. USMLE will require familiarity with only those basic techniques that have been covered here (although there arc many others). Their uses will now be summarized, as illustrated in Table 4-1.
Concerning nominal scale data, only one kind of question has been discussed: do the proportions of observations falling in different categories differ significantly from the proportions that would be expected by chance? The technique for such questions is the chi-square test.
Regarding ordinal scale data, only one kind of question has been mentioned: is there an association between ordinal position on one ranking and ordinal position on another ranking? The appropriate technique here is the Spearman rank-order correlation.
For interval or ratio scale data, three general kinds of questions have been discussed:
1. Questions concerning means:
• What is the tnte mean of the population?
• Is one sample mean significantly different from one or more other sample means?
2. Questions concerning variances:
• Are the variances in two samples significantly different?
3. Questions concerning association:
SCALE OF DATA Nominal Ordinal Interval or Ratio
U Differences in ¡r proportion
One or two means
S More than two means
P Variances N
E Association R
N Predicting the 1 value of a N variable G
(or z-test if n > 100)
ANOVA with F-tests
Three ways of answering questions concerning means of interval or ratio scale data have been examined: t-tests, z-tests, and ANOVA:
• When the question involves only one or two means, or making only one comparison, a t-test will normally be used. Therefore, questions concerning estimating a population mean, testing a hypothesis about a population mean, or comparing two sample means with each other will normally be answered by using t. Alternatively, provided that n > 100, or if the standard deviation of the population is known, a 7-test may be used with virtually identical results.
• When the question involves more than two means, or making more than one comparison, the appropriate technique is analysis of variance (ANOVA), together with F-tests.
One way of answering questions about variances has been covered: the F-test of significant differences between variances.
Two ways of assessing the degree of association between rwo interval or ratio scale variables have been discussed. To evaluate the strength and direction of the relationship, Pearson product-moment correlation is used, together with a form of t-test to test the null hypothesis that the relationship does not exist in the population. To make predictions about the value of one variable on the basis of the other, regression techniques are used.
Table 4-1 summarizes the range of inferential and correlational techniques that have ^ecn covered. This table should be memorized to answer typical USMLE questions that u require choosing the correct test or technique for a given research situation.
Select the single, best answer to the following questions.
1. A medical school professor finds that students' final examination grades correlate with the number of times they attended class, Pearson r = 0.8, p = .001. This means that a. a student will improve his or her grade by attending class more.
b. 64% of the variation in final grades is accounted for by class attendance.
c. the correlation is too low to be of significance.
d. the correlation is a weak one.
e. the correlation is nonlinear.
2. A lecturer states that the correlation coefficient between prefrontal blood flow under cognitive load and the severity of psychotic symptoms in schizophrenic patients is — 1.24. You can therefore conclude that a. prefrontal blood flow under cognitive load is a good predictor of the severity of psychotic symptoms in schizophrenic patients.
b. prefrontal blood flow under cognitive load accounts for a large proportion of the variance in psychotic symptoms in schizophrenic patienrs.
c. low prefrontal blood flow is in some way a cause or partial cause of psychosis.
d. psychosis or schizophrenia is in some way a cause or partial cause of low prefrontal blood flow under cognitive load.
e. the lecturer has reported the correlation coefficient incorrectly.
3. An investigator into the life expectancy of IV drug abusers divides a sample of patients into HIV-positive and HIV-negative groups. What type of data does this division constitute?
a. Nominal b. Ordinal c. Interval d. Ratio e. Continuous
4. The investigator in Question 3 finds that 169 of 212 HIV-positive IV drug abusers are no longer alive after 5 years, while only 64 of 439 HIV-negative IV drug abusers have died during this time. What statistical technique should he use to test the null hypothesis that there is no difference between these proportions?
a. t-test b. Correlation with associated t-test c. Chi-square d. Analysis of variance (ANOVA)
5. A researcher wishes to compare the effects of four different antiretroviral drug combinations on the survival time of two groups of patients with AIDS; one group are IV drug abusers, the other arc infants infected in utero. Each of these groups is divided into four subgroups; each subgroup is given a different drug combination. Which statistical technique would be most appropriate for analyzing the results of this study?
a. Analysis of variance (ANOVA)
b. t-test c. F-test d. Correlation with associated t-test e. Chi-square
6. A researcher claims that USMLE Step 1 scorcs can be predicted using the following equation:
where X, _ student's IQ, X2 = number of hours of daily study for the past year, and X3 = student's GPA at medical school. What statistical technique did the researcher use to arrive at this equation?
a. Spearman rank-order correlation b. Analysis of variance (ANOVA)
c. Regression d. Chi-square e. t-test
7. A study finds that there is a correlation of +0.7 between self-reported work satisfaction and life expectancy in a random sample of 5000 Americans (/> — 0.01). This means that a. work satisfaction is one factor involved in increasing one's life expectancy.
b. there is a strong statistically significant positive association between work satisfaction and life expectancy.
c. 70% of people who enjoy their work have an above-average life expectancy.
d. to live longer, one should try to enjoy one's work.
e. 70% of the variability in life expectancy in this sample can be accounted for by work satisfaction.
8. In a sample of 200 patients with hypertension who are currently taking antihypertensive medication, it is found that blood pressure and antihypertensive drug dosage correlate r = —0.3, p < .05. It is correct to conclude all of the following EXCEPT
a. the relationship between drug dosage and blood pressure is unlikely to be due to chance.
b. the relationship between drug dosage and blood pressure is a weak negative one.
c. although other factors are clearly involved also, drug dosage is one factor causing these patients' blood pressures to be reduced.
d. drug dosage accounts for 9% of the variation in blood pressures.
e. it would be possible to make a prediction of a patient's blood pressure from knowledge of their drug dosage by using regression techniques.
9. A study investigates two new drugs that are hypothesized to improve the mean level of recall in patients with Alzheimer's disease. A sample of 1000 patients (500 males, 500 females) arc ran domly allocated to receive Drug A, Drug B, or a placebo. After 3 months of treatment, all the patients' recall ability is tested. Males' recall is improved by Drug A but is made worse by Drug B, while the converse is true for females. Overall, however, there is no difference between the three treatments or between the two genders. It would be correct to report a. no effects.
b. a drug X gender interaction.
c. a main effect of age.
d. a main effect of gender.
e. a drug X gender interaction and a main effect of gender.
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