Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

which are closest to the regression line at the bivariate mean, (X,Y).

19.4.2.3 Additional Evaluations — Four additional evaluation procedures, relegated to the Appendix, are confidence intervals for individual points and for the intercept, comparisons of two regression lines, and transformations of r.

19.5 Pragmatic Applications and Problems

Regression and correlation coefficients are among the most frequently used indexes in statistics and are probably the most commonly abused. This section is concerned with the valuable applications. Section 19.6 discusses the abuses.

19.5.1“Screening” Tests for Trend

Correlation and regression coefficients are commonly used in medical literature as a “screening” (or sometimes definitive) test for whether two variables are related and for the average magnitude of the trend. If the variables seem to be distinctly related, the investigator can then speculate or do additional research about reasons for the relationship.

19.5.1.1 “Physical Examination” for “Abnormalities” — Before any statistical conclusions are formed from r, b, or P values, the graphical portrait of points should always receive a “physical examination.” The goal is to find “abnormalities,” such as distinctly curved patterns or major reversals in trend, that may make a straight-line inappropriate for the data. If the points are diffusely spread with no obvious pattern as in Figures 18.8 and 18.16, the rectilinear analysis is satisfactory for identifying

© 2002 by Chapman & Hall/CRC

an average trend. In such situations, a poor fitting line may be useless for accurate estimates, but can be valuable for denoting the average trend in the data, particularly if the analyst does not compare results in the zones, as suggested in Chapter 18. If the graph shows an obvious curving pattern, however, as in Figures 18.13 and 18.14, a rectilinear analysis may yield distorted results.

Figure 19.3 gives an excellent example of a published “screening process,” which was especially easy to do in this instance7 because the graph has points on both the X and Y sides of a (0,0) central axis. The points do not show an obviously curved pattern; and the straight line seems appropriate, with about equal numbers of points above it and below it throughout the graph.

FIGURE 19.3

Correlation between the vitamin K-induced changes in Ca2+ and hydroxyproline excretion. The straight line was fitted with a computer program equipped for linear regression calculations and the r value was + 0.437. [Figure and legend taken from Chapter Reference 7.]

After checking that a straight-line is suitable, the analyst can easily determine trend from the calculated value of r, which is readily obtained from computer programs. Many investigators prefer to cite b as the main quantitative index of the dependent relationship, but r has the advantage of being standardized, relatively unaffected by arbitrary units of measurement, and easy to interpret.

19.5.1.2 “Physical Diagnoses” of b or Wrong Lines — Skillful physical examiners can sometimes make good visual guesses of b without any calculations. One approach is to locate (and determine the slope of) a line that will have about equal numbers of points above and below it, consistent

axes have unequal-sized increments produced by logarithmic or other transformations, or if the axes have been truncated or changed in mid-course. Such axes, however, also provide opportunities for the artist to place graphic lines improperly and for the error to be diagnosed with simple physical examination.

For example, visual inspection would suggest that the solid line for Group A has been calculated or drawn incorrectly in Figure 19.4, which shows results of a clinical trial8 intended to find the effects of recombitant human erythropoietin on the packed cell volume (PCV) and serum erythropoietin (s-EPO) in patients donating blood before elective surgery. The solid squares are distributed so that about 32 of them are above the corresponding solid line and only about 11 are below that line.

Furthermore, the corresponding regression line seems to have a wrong intercept. In the line, which is reported to be log (s-EPO) = −.04(PCV) + 2.72, the customary Y-intercept cannot be checked because neither axis reaches zero, but the graph shows the X-intercept, where log s-EPO appears to be 0 when PCV is about 42.5. Inserting the latter value into the stated regression equation produces the disagreeing result that for PCV = 42.5, log s–EPO = (−.04)(42.5) + 2.72 = 1.02, which is about 1. Therefore,

© 2002 by Chapman & Hall/CRC

FIGURE 19.4

Relation between log s-EPO and PCV after surgery. Regression line equations: group A ( ), log s-EPO = –0.04 × PCV

(%) + 2.72 (r = − 0.76. p = 0.01); all r-HuEPO-treated patients ( ), log s-EPO = − 0.01 × PCV (%) + 1.82 (r = −0.38. p = 0.01). [Figure and legend taken from Chapter Reference 8.]

something must be wrong. (The problem may arise from the artist’s forgetting that the Y axis has a break between 0 and 1.)

A quick, crude way to check the value of b is to determine the range of X and Y dimensions as

(Xmax – Xmin) and (Ymax − Ymin) and then to estimate b as the ratio of (range of Y)/(range of X). For example, the solid squares in Figure 19.4 have a Y range from about 1.1 to 2, and an X-range from

about 24 to 40. The crude slope would be −0.9/16 = –.06, which is not far from the stated value of −.04.

19.5.1.3 Taxonomy of Visual Patterns for r — Even with advanced skill in physical diagnosis of graphs, most examiners have difficulty anticipating the values of r or r2. The difficulty arises from having to guess not just b, but also variances in the X and Y axes. Values of r near 0 or 1 are relatively easy to anticipate if the points have mainly a horizontal spread or a pattern that clearly looks like a straight line, but visual “diagnosis” is difficult for the intermediate zones between r = .2 and r = .8.

A collection of 10 graphs, excerpted from published medical literature, has been assembled here in Figures 19.5, 19.6, and 19.7, producing a taxonomy of visual patterns for r values ranging from near 0 to near 1. Visual guesses for r seem relatively easy for the extreme values near 0 and 1, but the zones between .2 and .8 do not have any obvious quantitative “diagnostic” signs.

Perhaps the main take-home message here is to be wary of any specifically quantitative claims based on regression and correlation coefficients. If r is big enough and stable, it supports the idea that the two variables are related on average. If the investigator claims they are closely or linearly related, however, and if the coefficient for b is offered as a confident prediction of future values, beware of the often contradictory graphical evidence.

© 2002 by Chapman & Hall/CRC

19.5.2Predictive Equations

ˆ

Although developed to allow Yi to be estimated from X i, the equation Yi = a + bXi rarely has a close enough fit to be used for individual predictive estimates in most medical research. Such estimates might be attempted when r ≥ .9, but few analysts would be willing to make specific predictions from the extensive variability shown, despite “r = .94,” in the lower right graph of Figure 19.7.

Predictive equations are regularly used for exact estimates, however, when diverse (usually chemical) substances are measured with laboratory technology. The magnitude of the measured substance (such

© 2002 by Chapman & Hall/CRC

FIGURE 19.6

Two graphs showing r values from .51 to .60.

as a concentration of serum calcium) is usually converted into the magnitude of another entity (such as the voltage on a spectrophotometer), and a “calibration curve” is constructed, as shown in Figure 19.8, for the voltage values at known magnitudes of the substance. If the calibration points produce a closely fitting straight line, the equation is then used to estimate the values of future “unknown” substances. For pragmatic application, the equation has a reverse biologic orientation. Biologically, the voltage readings depend on the concentrations in the serum, but pragmatically, future concentrations will be estimated from the voltage readings. Accordingly, the voltages are plotted as the X variable and concentrations as Y.

Equations of this type are invaluable in modern laboratories, but are misapplied if the goal is to compare measurements of the same substance, such as serum calcium, by two different methods, such as flame photometry vs. spectrophotometry. If the analytic aim is to determine agreement rather than trend, the best approach is a concordance method described in Chapter 20.

© 2002 by Chapman & Hall/CRC

19.5.3Other Applications

Although screening for trend and making predictive estimates are the most common applications, regression/correlation procedures can be applied for some of the other goals noted in the next few subsections.

© 2002 by Chapman & Hall/CRC

19.5.3.1Summarizing Individual Time Trends — Each person’s individual data (in a group of people) may be a series of measurements over a period of time. For example, the response to cholesterol-lowering treatment may comprise serum cholesterol measurements at baseline and at four time intervals thereafter. To summarize individual results, a regression equation can be fitted for each person’s trend over time; and the individual regression coefficients can represent the main “outcome” variable.

19.5.3.2“Smoothing” Vital Statistics Rates — The annual rates of incidence (or prevalence) in vital statistics phenomena (such as birth, death, or occurrence of disease) often show oscillations from one year to the next. To avoid these oscillations, the analysts may try to “smooth the curve” by using a “running average,” in which each year’s value is calculated as a mean of values for the current, preceding, and following year. Thus, the incidence rate for 1992 might be an average of the rates for 1991, 1992, and 1993. In an alternative approach, a lengthy (or the entire) sequence of time points is “smoothed” by conversion to a regression line.

19.5.3.3Curvilinear and Quadratic Regression — If the plot of points is obviously curved,

= a + b(1/Xi ).

In other situations, however, X may be retained without transformation, but expressed in a frankly curvilinear polynomial such as

The most common polynomial format, called quadratic regression, adds only one extra quadratic term,

A quick screening “diagnosis” for the shape of the relationship can be shown by the ratio of the standard deviation (or its square) to the mean of the subset of Yi values in equally spaced intervals of Xi. The regression model is linear if this ratio remains about constant, and log linear (i.e.,ogl Y = a + bX) if the ratio progressively increases. If the square of the standard deviation increases in proportion to the mean, the regression model is quadratic (i.e., Y = a + bX + cX2).

19.5.3.4 Time Series and Financial Applications — Some bivariate analyses are intended to show the “secular” changes during calendar time of a dependent variable, such as the population or economic status of a geographic region. The statistical goal might be satisfied with a best-fitting straight line for population size as Y variable and for time as the X variable. The patterns of data, however, may form a curve with wavy or irregular shapes that are not well fitted with a straight line; or the data analyst may want to make particularly accurate predictions for the future, based on what was noted in the past.

The association may therefore be described not with a crude straight-line model, but with a polynomial or other complicated algebraic expression called a time series. The relationship is still bivariate, but the

relationships are commonly used in economics to help predict stock-market prices or in meteorology to forecast weather. Having seldom been usefully employed in clinical or epidemiologic activities, timeseries analysis will not be further discussed here.

In financial enclaves, an ordinary linear regression model, Y = a + bX, is sometimes used to express the performance of individual stocks during a period of time, such as a month or year. In this model, Y represents daily prices of the stock and X represents the corresponding values of a market index, such as the Dow-Jones average. The b coefficient, called beta, indicates the stock’s movement in relation to the market (or “how much the market drives the stock”), but the a intercept, called alpha, indicates unique attributes of the stock itself. According to the theory, alpha should equal zero on average, but stocks with a high positive alpha are believed to have a high likelihood of going up rapidly in price. (The standard deviation of the Y values is sometimes used as an index of “volatility.”)

© 2002 by Chapman & Hall/CRC

19.6 Common Abuses

The four most common types of abuse for correlation and/or regression procedures are: (1) drawing “significant” conclusions from stochastic P values, while ignoring the actual quantitative significance of the relationship; (2) failure to check whether a straight-line model is suitable for the actual shape of the relationship; (3) ignoring the potential influence of outliers; and (4) concluding that a causal relationship has been shown when high values of correlation indicate a strong association.

19.6.1Stochastic Distortions

Formula [19.20] for the stochastic t value can be re-written as

In this formula, the r/ 1 – r2 ratio is analogous to a standardized increment. Its changing value for different values of r is shown in Table 19.2. On the left side of the table, with small values of r (below

.4), the value of 1 − r2 and the enlarged 1 – r2 will be close to 1, and so the ratio of r/( 1 – r2 ) will be essentially r. On the right side of the table, as r enlarges above .5, the ratio becomes strikingly larger than r.

TABLE 19.2

Values of r, 1− r2, and r/ 1 – r2

To get across the stochastic boundary for “significance” at P ð .05, t must be at least 2. (The requirement is somewhat higher with small group sizes.) Because the ratio exceeds 2 when r ≥ .9, this magnitude of r will almost always be stochastically significant. At levels of r ≥ .5, relatively small values of n will make the product high enough when the ratio of r/ 1 – r2 is multiplied by n – 2 . The most striking effect occurs at small values of r, however, where sufficiently large group sizes for n will always produce a t value that exceeds 2. The most trivial value of r can thus become stochastically significant if n is big enough.

For example, with a negligibly small r value of .01, the P value will be < .05 if the sample size is

© 2002 by Chapman & Hall/CRC

Formula [19.25] can promptly show the group sizes that will transform r values into stochastic significance. Using t = 2 as a rough guide, the required value will be n = [4(l – r2)/r2] + 2. Table 19.3 shows the values of n needed to get t ≥ 2 for different values of r. For example, a correlation coefficient of .7, which has reduced less than half of the original group variance (r2 = .49), will be stochastically significant if obtained with as few as 7 people. (One more person will actually be required here because t must exceed 2 as group sizes become very small. In this instance, t would have to be about 2.37, rather than 2; and Formula [19.25] would produce n = 7.85 rather than 7.) An r value of .2, which has reduced only 4% of the original variance, will be stochastically significant if the group size exceeds 98.

TABLE 19.3

Values of n Required for Stochastic Significance (P ð .05) at Cited Value of r

For this reason, the associated P value is the worst way to determine whether a correlation coefficient is quantitatively significant. Nevertheless, either to avoid making a scientific judgment about the magnitude of r or to escape the distressing implications of low values for r (or r2 ), many investigators use only a stochastic calculation of P to claim that an association is “significant.” This pernicious abuse of statistics is abetted by reviewers and editors who allow the authors to “get away with it.”

in the midst of technologic onslaughts, was reproached, as the contemporary editor of Circulation, by the authors10 of a paper in which “information regarding statistical significance of certain [r correlation coefficients] ... was deleted.” Stead defended11 the editorial deletion by saying that “the P value is frequently misleading [and that] … when the number of observations is large ... a relationship between two variables at the mathematical level does not necessarily translate into a useful relationship at the patient care level.”

This type of intellectual vigor seems to be disappearing among editors today. For example, in Figure 19.9, which is reproduced exactly as published,12 the investigators showed “three data points” for each of “63 men … who had stable angina pectoris … and a positive exercise test with ST-segment changes indicative of ischemia.” Without citing a value of r or r2, the authors concluded that “For this range of carboxyhemoglobin values, there was a decrease of approximately 3.9 percent in the length of time to the ST end point for every 1 percent increase in the carboxyhemoglobin level.”

You can decide for yourself whether the claimed precision of the linear relationship is really supported by the points in Figure 19.9. The decision can be helped by checking in a table of t/P values to find that the highly impressive “P ≤ 0.0001” requires a value of t ≥ 4. Inserting n = 3 × 63 = 169 into Formula [19.25] produces 169 = [16(1 − r2)/r2] + 2, which can be solved to show that 11.44r2 = 1. Consequently, an r 2 value as small as 1/11.44, or about .09, could be associated with the “impressive” findings and subsequent quantitative precision of the relationship claimed by the investigators.

The abuse of P values is now so thoroughly entrenched that many published reports do not even show values of r, b, or the regression equation. They are replaced entirely by P values. The P values themselves are also sometimes replaced by “stars” (usually asterisks) in which * represents < .05, ** is < .01, and

***is < .001. The starry results, showing neither descriptive values nor the actual P values, may then appear in a celestial table that is bereft of any scientific evidence of quantitative effect. [The use of stars to indicate P values seems to have been introduced,* in a textbook discussion of regression methods, by the distinguished statisticians, G.W. Snedecor and W.G. Cochran.13]

Because P values are so affected by group size, they should never alone be trusted for decisions about

“significance.” A sole reliance on P values may also substantially distort decisions about “nonsignificance.” If the P value is >.05 because only four patients were studied, a strong correlation coefficient,

as high as .8, might be dismissed as “nonsignificant” (without confirmation by a confidence interval).

*The introduction probably came in the 5th edition in 1956, as the tactic is not mentioned in the 1st edition (written by

Snedecor alone) in 1937, or in the 4th edition in 1946.

© 2002 by Chapman & Hall/CRC

Dose-Response Relation between the Percentage Change in the Length of Time of the Threshold Ischemic ST-Segment Change (ST End Point) and the Carboxyhemoglobin (COHb) Level after Exercise. Each subject is represented by three data points. The regression shown represents the mean slope of the individual regressions for each subject when y = 3.9 (percentage of COHb) + 8.0 (P ð 0.0001). [Figure and legend taken from Ch apter Reference 12.]

19.6.2Checks for Linear Nonconformity of Quantified Trends

When b and r are determined, the bivariate pattern is fitted with a straight line whose slope is constant and unchanging throughout all zones of the data. Nevertheless, as shown in Chapter 18, some data sets will not be suitably analyzed with a straight line. For other data sets, as in the right side of Figure 18.13, a straight line is obviously the wrong model. It will produce a fallacious result, with r and b near 0, suggesting no relationship, although the two variables actually have a close parabolic relationship. In yet other situations, as in the middle of Figure 18.13, a straight line will produce a reasonably good fit, but the slope of the line will be misleading. It will be too high in the lower and upper ends of the data and too low in the center. (In other situations, such as Figure 19.9, the appropriate regression line may not be curved, but may fit so poorly that any claims for a precise relationship must be doubted.)

19.6.2.1 Quantitative Impact of “Risk Factors” — Checking the fit of the line becomes important when a quantified trend in “risk factors” is claimed on the basis of coefficients obtained with various forms of regression analyses. The customary statement is something like “You gain 3.4 years of life for every fall of 5.6 units in substance X.” Because the claims usually come from data in which the outcome event is a 0/1 binary variable for life or death, the regression lines (or “surfaces” in multivariable analysis) cannot be expected to have the close fit that might be anticipated for dimensional measurements of biochemical or physiologic variables.

Therefore, before any quantitative contentions, the analyst should first determine whether the data have curved patterns and whether a rectilinear regression model is appropriate. As noted by Concato et al.14, however, many published quantitative claims have not been checked for linear conformity between the regression model and the data.

19.6.2.2 Plots of Residual Values vs. Xi — A conventional mathematical method of doing

ˆ

this check is to examine the plot of the residual values, Yi – Yi , against X i . If a straight line gives a proper fit, the plot of residuals should show points that vary randomly around 0 as X goes from its

© 2002 by Chapman & Hall/CRC