Models to Assess the Association of a Semiquantitative Exposure With Outcomes

Abstract

A semiquantitative risk factor has 2 components: any exposure (yes/no) and the quantitative amount of exposure (if exposed). We describe the statistical properties of alternative analyses with such a risk factor using linear, logistic, or Cox proportional hazards models. Often analyses employ the amount exposed as a single quantitative covariate, including the nonexposed with value zero. However, this analysis provides a biased estimate of the exposure coefficient (slope) and we describe the magnitude of the bias. This bias can be eliminated by adding a binary covariate for exposed versus not to the model. This 2-factor analysis captures the full risk-factor effect on the outcome. However, the coefficient for any exposure versus not does not have a meaningful interpretation. Alternatively, when exposure values among those exposed are centered (by subtracting the mean), the estimate of this coefficient represents the difference in the outcome between those exposed versus not in aggregate. We also show that the biased model provides biased estimates of the coefficients for other covariates added to the model. Proper analysis of a semiquantitative risk factor should start with a 2-factor model, with centering, to assess the joint contributions of the 2 components of the risk-factor exposure. Properties of models were illustrated using data from a multisite study in North America (1983–2019).

Abbreviations

bpm

beats per minute

CCN

confirmed clinical neuropathy

df

degrees of freedom

EDIC

Epidemiology of Diabetes Interventions and Complications

LDL

low-density lipoprotein

Epidemiology research often employs a semiquantitative exposure (e.g., smoking) that is actually a mixture of a binary covariate for exposure (yes/no) and a quantitative covariate for the amount of exposure among those exposed. Methods for the analysis of such data as the outcome or dependent variable have been described (1, 2), including where the outcome variable is truncated from below (3). Herein, however, we use a semiquantitative variable as a risk factor in a regression model to assess the association of any exposure (e.g., smoking yes/no) and the exposure level (e.g., smoking intensity) with an outcome.

Often the analysis has employed a single quantitative covariate, including the nonexposed subjects with an intensity value of zero. For example, Feodoroff et al. (4) recently presented an analysis of the association of smoking with the risk of cardiovascular disease in a large cohort of subjects with type I (insulin-dependent) diabetes. Their models included separate coefficients for the smoking intensity (pack-years smoking) among current smokers and former smokers, and included the nonsmokers with value zero. However, Leffondré et al. (5), in an example (i.e., numerical computation), suggested that such analyses that do not account for the difference between smokers and nonsmokers provide a biased estimate of the coefficient or slope of the outcome regressed on the amount exposed. Herein we provide a derivation of the statistical expression for the magnitude of the bias. We also show that such analyses that include other covariates provide biased estimates of the coefficients for those covariates as well.

In some analyses the quantitative component of the exposure has been centered by subtracting the mean from each value, and the nonexposed have been included with the exposure amount zero (5–7). However, the statistical properties of a centered model have not been described. Such data have also been used to estimate a nonlinear exposure-response curve (8, 9), focusing on the goodness of fit of the nonlinear model.

Herein we describe the interpretation of the corresponding coefficients in noncentered and mean-centered linear models for a quantitative outcome, a logistic regression model for a binary outcome, and a Cox proportional hazards model for an event-time outcome. We show that a model with the quantitative exposure alone, that also includes the nonexposed with exposure value zero, provides a biased estimate of the exposure coefficient relative to that in a model restricted to those exposed. Further, if other factors are added to this model, their coefficient estimates are also biased relative to a model that includes a binary variable for those exposed versus not. Thus, preferable is a 2-factor model with centered exposure values that yields unbiased estimates of the model coefficients and those of other covariates added to the model, and provides an overall comparison of the outcome for those exposed versus not in addition to the association with the amount of exposure. We have illustrated properties of models with application to data from the Diabetes Control and Complications Trial and Epidemiology of Diabetes Interventions and Complications study (DCCT/EDIC) (clinical trials registration numbers: NCT00360893, NCT00360815), conducted in the United States and Canada. These studies span the period from 1983 to date.

LINEAR MODELS FOR A SEMIQUANTITATIVE RISK FACTOR

Suppose that a semiquantitative exposure variable is employed as a risk factor (predictor) for a quantitative outcome in a simple linear regression model. Let the binary variable designate whether a subject was exposed versus not , and be the amount of exposure in the of the observations that were exposed, where if not exposed.

As examples, we have conducted analyses of smoking exposure among participants in the study of the Epidemiology of Diabetes Interventions and Complications (EDIC), an observational study of the incidence and prevalence of micro- and macrovascular complications in over 1,000 participants with type 1 diabetes mellitus, followed annually (10). Smokers included current smokers and those who quit within 3 months prior to an annual visit, and the intensity is the current average number of cigarettes smoked per day. Nonsmokers did not currently smoke or previously smoked and quit 3 or more months prior to the visit.

Consider models to assess the association of smoking with pulse rate (beats per minute (bpm)) after 2-years of follow-up in EDIC. Among the 268 smokers, the mean intensity was 15.97 cigarettes per day and the mean pulse rate 75.42 bpm, whereas for the 1,012 nonsmokers the mean intensity was 0 by definition and the mean pulse rate was 72.11 bpm. The mean difference in pulse rate among smokers versus nonsmokers is , with standard error = 0.7113 yielding t = 4.65 and P < 0.0001. However, this ignores the potential influence of smoking intensity.

Table 1 presents the coefficient estimates for additional models for the EDIC pulse rate data.

Table 1.

Linear Regression Models of the Association of the Prevalence of Smoking Versus Not and the Smoking Intensity ( or ) With Pulse Rate (Beats per Minute, ) at 2 Years of Follow-up in the Epidemiology of Diabetes Interventions and Complications Cohort, North America, Circa 1995

Model	Coefficient   (Factor)	Estimate	SE	P  Value	Model F	df	P  Value
Noncentereda 1:		75.1046	1.1994	<0.0001	0.098	1	0.7546
		0.0199	0.0634	0.7546
Noncentereda 2:		72.3224	0.3145	<0.0001	15.73	1	<0.0001
		0.1442	0.0364	<0.0001
Noncentereda 3:		72.1117	0.3256	<0.0001	10.87	2	<0.0001
		0.0199	0.0626	0.7511
		2.9929	1.2268	0.0148
Centeredb 4:		75.4216	0.6415	<0.0001	0.098	1	0.7546
		0.0199	0.0634	0.7546
Centeredb 5:		72.8047	0.2918	<0.0001	0.099	1	0.7530
		0.0199	0.0631	0.7530
Centeredb 6:		72.1117	0.3256	<0.0001	10.87	2	<0.0001
		0.0199	0.0626	0.7511
		3.3100	0.7115	<0.0001

Abbreviations: df, degrees of freedom; SE, standard error.

^a Models with noncentered smoking intensities (x, models 1–3).

^b Models with centered intensities (c, after subtracting the mean, models 4–6).

Semiquantitative variable without centering

The quantitative association of smoking intensity with pulse rate among the smokers can be directly assessed using model 1:

(1)

where the errors are assumed to have mean zero and constant variance for all observations. The coefficient provides an unbiased estimate of the slope of pulse rate on smoking intensity—in this case not significant, with and P = 0.7546. However, this does not allow for a general association with smoking (i.e., a generally higher pulse rate among smokers vs. nonsmokers independent of the amount smoked). Also, under this model the intercept is the estimate of the mean of among smokers who smoked 0 cigarettes per day , which is a hypothetical, nonexistent value given that there are no smokers with intensity .

Often, however, the analysis employs all subjects, including nonsmokers, as in model 2:

(2)

where for nonsmokers. This model shows a significant association of the outcome with smoking in all 1,280 subjects including the 1,012 nonsmokers with (P < 0.0001). However, Web Appendix 1 shows that the coefficient estimates and are biased relative to the coefficients and in the unbiased model 1, the slope estimate being relative to . Thus, the model estimates the pulse rate to be 0.1442-bpm higher for a smoker of 1 cigarette per day versus a nonsmoker, as well as 0.1442-bpm higher for a smoker of cigarettes per day versus a smoker of cigarettes per day for any . This is implausible.

In addition, the standard error of the intensity coefficient in this biased model 2 is about half that in the unbiased model 3 (0.0364 versus 0.0634) with resulting P values of < 0.0001 versus 0.7546. This would lead to the erroneous conclusion that the smoking intensity in this case is positively associated with the pulse rate.

Alternatively, a 2-factor mixture model would assess both the association of smoking versus not and the smoking intensity with the outcome , as in model 3:

(3)

with coefficient vector . Using least-squares estimation the coefficient estimates satisfy

(4)

so that the intercept is the mean of among nonsmokers, is the slope of on among smokers as in model 1, and is the difference between the intercept from model 1 and the mean of among nonsmokers. While the slope is the same as in model 1, the standard error, , and P are slightly different owing to the addition of the binary component with estimate , P = 0.0148. The model test of the joint null hypothesis for the full influence of smoking is highly significant (F = 10.87 on 2 degrees of freedom (df), P < 0.0001), more so than either 1-df test of each component.

Figure 1 then depicts the relationships in model 3. The estimated intercept is , which equals the mean pulse among nonsmokers . The estimated regression line provides the estimated mean pulse rate over the range of intensity values among smokers computed as . Given that and the regression line is computed as for .

Figure 1.

Estimated mean pulse (beats per minute (bpm), ) from the noncentered model 3 in (equation 3) as a function of the smoking intensity where designates the nonsmoker subset of the cohort and designates the smoking intensity among smokers, using data from the Epidemiology of Diabetes Interventions and Complication study, North America, circa 1995. The point marked by the triangle is the model-estimated intercept of the smokers-only model 1 in (equation 1), the diamond is the model-estimated pulse rate for a smoker with intensity = 1, the square is the mean pulse rate among smokers at the average intensity, and the circle is the mean pulse rate among nonsmokers . The model 3 estimate of the coefficient for smoker versus not in (equation 3) equals the difference in pulse rate for points triangle minus circle. This figure was generated by plotting a point at (0, 72.1117) and then a line connecting the points (, ), .

Note that the test of the hypothesis represents a test of the hypothesis that the mean of among the nonsmokers equals the intercept of the quantitative association alone or the difference between the points marked by the triangle and circle in the figure, and is not very meaningful. Rather, it would be of greater interest to test the difference in the mean pulse between smokers versus nonsmokers (i.e., the vertical difference between the points marked by the square and circle).

In summary, model 1 shows that the slope of pulse rate on the smoking intensity among smokers is not significantly different from a horizontal line (slope = 0). Model 2 shows that adding the 1,012 nonsmokers to the intensity data set yields a highly significant slope estimate (, P < 0.0001) but this estimate is severely biased owing to the discontinuity between the regression line among smokers and the lower average pulse rate among nonsmokers. Model 3 shows that inclusion of the nonsmokers with an additional term in the model for smoker versus not is highly significant (P < 0.0001, 2-df) with the coefficient for smoker versus not being with P = 0.0148. However, this coefficient does not provide an estimate of the overall difference in the mean pulse rate among smokers versus nonsmokers.

Semiquantitative variable with centering

Often the smoking intensity (or amount of exposure) is centered by subtracting the mean intensity among smokers from the smoking intensity values, but retaining the value for nonsmokers. Thus, is transformed to a centered semiquantitative variable, say , as

(5)

Table 1 shows the centered models. The equivalent of model 1 is model 4:

(6)

where e = 1, using only the smokers. Then, is the estimated mean pulse associated with , or with the mean intensity . This can be obtained as . Also, from the noncentered model 1 and is nonsignificant.

The centered equivalent to model 2 using all observations including nonsmokers is model 5:

(7)

Unlike model 2, which provided a highly significant biased estimate of the slope, model 5 provides an unbiased estimate of the slope that is nonsignificant and virtually identical to that of model 4. Further, Web Appendix 2 shows that model 5 provides an unbiased estimate of the slope when the 2-factor model is correct. Thus, including nonexposed subjects with value zero in addition to the mean-centered data for those exposed also yields an unbiased estimate of the slope .The centered 2-factor model is model 6:

(8)

where it can then be shown that the coefficient estimates satisfy

(9)

The noncentered 2-factor model 3 and centered model 6 yield identical model F tests, intercepts and slopes, standard errors, and P values. However, this centered model provides a more relevant representation of the association of smoking versus not with a coefficient estimate that is the difference in the mean pulse rate among smokers versus that among nonsmokers . Conversely, the coefficient from the noncentered model 3 is the difference between the intercept of the smokers-only model 1 and the mean pulse rate among nonsmokers . Thus, the tests of the hypotheses and differ, the latter being more relevant, and more significant, in these data.

Web Appendix 2 describes the expected values of the coefficient estimates for these 3 centered models and shows that the coefficient estimate for the quantitative exposure is unbiased in each.

Figure 2 then describes these results. Using model 6, the 2 components of smoking have a significant model F test, with P < 0.0001. Among smokers the pulse rate is expected to increase by 0.0199 bpm per cigarette smoked, which is not significantly different from zero. However, the mean pulse rate between smokers and nonsmokers differs by bpm, which is highly statistically significant.

Figure 2.

Estimated mean pulse rate (beats per minute (bpm), ) from the centered model 6 in (equation 8) as a function of the centered smoking intensity among smokers and nonsmokers , using data from the Epidemiology of Diabetes Interventions and Complication study, North America, circa 1995. The regression line starts from the centered intensity equivalent to an intensity of 0 (the point marked by the diamond = − = −15.23) where the square point is the intersection of the mean pulse and the mean centered intensity (= 0) among smokers, and the circle is the mean pulse rate among nonsmokers. The model (equation 8) coefficient estimate equals the difference in pulse rate for points square minus circle.

Other covariate associations

In a multivariate model, the coefficient estimate for each covariate is a function of the estimates for the other covariates. For example, consider a model with smoking (yes/no), smoking intensity, and low-density lipoprotein (LDL, in mg/dL) as predictors. Any bias in the estimates of the coefficients for the semiquantitative smoking variables will then introduce a bias in the coefficient estimate for LDL. An explicit proof is presented in Web Appendix 3.

For example, consider a model to assess the joint effects of smoking and LDL levels on pulse rate in the full cohort, not just for smokers. In this case, the addition of LDL to models 1 and 4 in smokers alone is not relevant. The LDL coefficient estimates per 10 mg/dL in a univariate (unadjusted) model, and when added to models 2, 3, 5 and 6 using all subjects, are shown in Table 2.

Table 2.

The Low-Density Lipoprotein Coefficient Estimates per 10 mg/dL in a Univariate (Unadjusted) Model and When Added to Models 2, 3, 5, and 6 (Including the Smoking Variables) With the Full Sample, Epidemiology of Diabetes Interventions and Complications Cohort, North America, Circa 1995

Model	LDL Estimate	SE	Value	P Value
LDL alone	0.510	0.094	5.43	<0.0001
LDL + smoking alone	0.481	0.094	5.14	<0.0001
2: LDL + noncentered intensity	0.480	0.094	5.12	<0.0001
3: LDL + noncentered 2-factor	0.481	0.094	5.13	<0.0001
5: LDL + centered intensity	0.510	0.094	5.42	<0.0001
6: LDL + centered 2-factor	0.481	0.094	5.13	<0.0001

Abbreviations: LDL, low-density lipoprotein; SE, standard error.

With no adjustment for smoking, the LDL coefficient in the full cohort is 0.510. Adjusting for smoking (or any other covariate that is correlated with LDL) is expected to provide a different coefficient value (see Web Appendix 3). Adjusting for smoker versus nonsmoker alone yields an LDL coefficient of 0.481. The same coefficient is provided by the unbiased 2-factor models 3 and 6. The noncentered 1-factor model 2, which provides a biased estimate of the coefficient for , also provides an unbiased estimate of the LDL coefficient. However, the centered, 1-factor model 5 provides a biased estimate of the adjusted LDL coefficient, being equal to that of the LDL alone with no adjustment. Thus, the centered model 5 corrects for the bias in the intensity slope in the uncentered model, but it introduces a bias in the LDL slope. Again the 2-factor model with LDL added would be preferred.

NONLINEAR REGRESSION MODELS

We now show that similar relationships apply to the logistic and Cox proportional hazards models. As is shown for linear models in Web Appendix 1, expressions for the expected values of the model coefficients can be obtained along the lines described in Emond et al. (11) and also show that the model 2 coefficients are biased.

Logistic regression model

Consider a logistic regression model where is a binary variable to indicate a positive versus negative response or outcome. The model then estimates the logit of the probability that given smoking intensity , designated as , where is the probability among nonsmokers . Thus, the 3 noncentered models are

(10)

and the 3 centered models 4–6 likewise as a function of parameters , , and , respectively.

For illustration consider the association of smoking at EDIC year 13 with the presence or absence of confirmed clinical neuropathy (CCN) at year 13. Among the 168 smokers, the mean intensity is 15.23 cigarettes per day and the prevalence of CCN is 65 subjects (38.7%) whereas for the 984 nonsmokers the mean intensity is 0 by definition and the prevalence is 281 subjects (28.6%). The odds ratio of CCN for smokers versus nonsmokers is 1.579 (P = 0.0084) ignoring the amount smoked.

Table 3 presents the coefficient estimates for each model. In model 1, using only the smokers , the coefficient relating the logit of the prevalence of CCN to the quantitative smoking intensity is not significant.

Table 3.

Logistic Regression Models of the Association of the Prevalence of Smoking Versus Not and the Smoking Intensity ( or ) With the Prevalence of Confirmed Clinical Neuropathy at 13 Years of Follow-up in the Epidemiology of Diabetes Interventions and Complications Cohort, North America, Circa 2006

Model	Coefficient   (Factor)	Estimate a	SE	P Value	Model χ 2	df	P Value
Noncenteredb 1:		−0.5891	0.3064	0.0545	0.24	1	0.6217
		0.0084	0.0170	0.6217
Noncenteredb 2:		−0.9010	0.0687	<0.0001	6.09	1	0.0136
		0.0232	0.0094	0.0136
Noncenteredb 3:		−0.9170	0.0706	<0.0001	7.19	2	0.0274
		0.0084	0.0170	0.6215
		0.3280	0.3144	0.2969
Centeredc 4:		−0.4610	0.1585	0.0036	0.24	1	0.6217
		0.0084	0.0170	0.6217
Centeredc 5:		−0.8459	0.0643	<0.0001	0.27	1	0.6001
		0.0094	0.0180	0.6001
Centeredc 6:		−0.9170	0.0706	<0.0001	7.19	2	0.0274
		0.0084	0.0170	0.6215
		0.4561	0.1735	0.0086

Abbreviations: df, degrees of freedom; SE, standard error.

^a Estimate is the log odds ratio.

^b Models with noncentered smoking intensities (x, models 1–3).

^c Models with centered intensities (c, after subtracting the mean, models 4–6).

Model 2 then expresses the probability of CCN as a function of alone among all 1,152 subjects, including nonsmokers, with coefficient estimate , P = 0.0136. However, as for the linear model, this coefficient estimate is biased, being almost 3-fold greater than the coefficient in model 1 owing to the inclusion of the 984 nonsmokers with intensity value zero.

The mixture model 3 intercept is or the logit of the proportion with CCN among the nonsmokers. As in the linear model, the smoking intensity coefficient is the same, . Then,   satisfies almost exactly , as in the case of a linear model. Although the 2-factor model test is significant, with P = 0.0274, neither of the 1-df tests of the coefficients for the 2 components or is significant, as might occasionally occur.

Figure 3 then shows the prevalence (probability CCN present) for nonsmokers and smokers with intensity over the range of levels of intensity . Let denote the inverse logit or logistic function . Under the 2-factor mixture model:

(11)

for .

Figure 3.

Probability of confirmed clinical neuropathy as a logistic function of the smoking intensity where designates the nonsmoker subset of the study cohort and designates the smoking intensity among smokers, using data at 13 years of follow-up from the Epidemiology of Diabetes Interventions and Complication study, North America, circa 2006. The figure plots the point marked by the circle = (0, ) and the line with points (, , .

Models 4 and 5 are nearly equivalent, as for the linear model (see Table 3). In model 6, given that the logit of the probability of CCN is the outcome, it follows that

(12)

which equals the log odds ratio of CCN for smokers versus nonsmokers. The model-based estimated logits of the probabilities are and with a difference that yields the estimate . A similar computation using the logits of the simple prevalences (0.3869 versus 0.2856) yields a trivially different log odds ratio of .

Then a test of is a test of the equality of the prevalence among smokers versus nonsmokers. In this example, the test of is significant whereas that of is not. As with the linear model, the coefficient provides a more meaningful description of the difference between smokers and nonsmokers than does in the noncentered model 3.

Cox proportional hazards model

Consider a Cox proportional hazards model of the association of smoking at EDIC year 2 with the time to a cardiovascular event or right censoring over 20 subsequent years of follow-up. Among the 268 smokers the mean intensity is 15.63 cigarettes per day, and 43 (16.0%) subjects were cardiovascular disease cases, whereas among the 1,026 nonsmokers the mean intensity is 0 by definition, and 107 (10.4%) were cases. The hazard ratio of cardiovascular disease for smokers versus nonsmokers is 1.63 (P = 0.007) ignoring the amount smoked.

The 3 noncentered models are:

(13)

and the 3 centered models 4–6 likewise as a function of parameters , and , respectively. Table 4 shows the estimated coefficients.

Table 4.

Cox Proportional Hazards Regression Models Relating the Prevalence of Smoking Versus Not and the Smoking Intensity ( or ) at 2 Years of Follow-up to the Risk (Hazard) of an Initial Cardiovascular Event in the Epidemiology of Diabetes Interventions and Complications Cohort, North America, Circa 2006

Model	Coefficient   (Factor)	Estimate a	SE	χ 2  Value	P Value	Model χ 2	df	P Value
Noncenteredb 1:		0.0109	0.0152	0.52	0.4726	0.52	1	0.4726
Noncenteredb 2		0.0234	0.0086	7.44	0.0064	7.44	1	0.0064
Noncenteredb 3		0.0112	0.0153	0.53	0.4657	8.08	2	0.0176
		0.3125	0.3099	1.02	0.3132
Centeredc 4:		0.0109	0.0152	0.52	0.4726	0.52	1	0.4726
Centeredc 5		0.0148	0.0180	0.68	0.4111	0.68	1	0.4111
Centeredc 6		0.0112	0.0153	0.53	0.4657	8.08	2	0.0176
		0.4869	0.1810	7.23	0.0072

Abbreviations: df, degrees of freedom; SE, Standard error.

^a Estimate is the log hazard ratio.

^b Models with noncentered smoking intensities (x, models 1–3).

^c Models with centered intensities (c, after subtracting the mean, models 4–6).

As in the other analyses, the coefficient for model 2 using all subjects yields a strongly biased overestimate of the smoking intensity coefficient relative to model 1. However, this bias is reduced but not eliminated in the centered model 5.

Because the background hazard functions for models 1 versus 3 differ, the coefficients for the smoking intensity and differ slightly. Centered models 4 and 6 show the same coefficient for smoking intensity as do the noncentered models 1 and 3.

In models 3 and 6, the 2-factor model χ²-test yields the value 8.08 with P = 0.0176. In model 3 the tests of and are both nonsignificant. However, in model 6, the test of is significant, and the coefficient yields a hazard ratio of for smokers versus nonsmokers.

Figure 4 then presents the hazard ratio using model 6 (13) as a function of the centered smoking intensity , the intercept having been absorbed into the background hazard.

Figure 4.

Hazard ratio for the risk of a cardiovascular event for smokers at a given centered smoking intensity (, where is the smoking intensity and = 15.63) versus the risk among nonsmokers as an exponential function of , using data at 13 years of follow-up from the Epidemiology of Diabetes Interventions and Complication study, North America, circa 2006. This line connects the points (, ).

DISCUSSION

Semiquantitative measures are often employed as risk factors in epidemiology where smoking and other exposures are common. Such variables are a mixture of a binary component (e, exposed versus not) and a quantitative component x for those exposed. Thus, statistical models should employ terms for both components to capture the full influence of the exposure variable.

However, a common approach is to use the exposure as a single covariate in a model with values zero for those not exposed (model 2 herein). For example, a recent article assessed the association of smoking intensity (packs per day) separately among current and former smokers with the risk of coronary heart disease outcomes using the Cox proportional hazards model. They reported that “One pack per day significantly increased the risk of incident coronary heart disease in current smokers compared with never smokers (HR 1.45 [95% CI 1.15, 1.84])” (4, p. 2580). However, this interpretation is incorrect. This is not the hazard ratio comparing current smokers with nonsmokers but rather an estimate of the hazard ratio of coronary heart disease per pack per day where the hazard ratio for a smoker of 1 pack per day versus a nonsmoker is assumed to be the same as the hazard ratio for a smoker of 2 versus 1 pack per day, or 11 versus 10, etc. This is implausible. Rather, one might expect that there is a fundamental difference in risk for smokers versus nonsmokers in addition to the difference in risk associated with an increase in the smoking intensity (i.e., that the 2-factor model is correct). In this case, the estimate of the coefficient for the smoking intensity (log hazard ratio per unit intensity) in a model with intensity as the only factor (with value zero for nonsmokers), that is, model 2 above, is biased compared with a model restricted to only include current smokers (model 1 above).

Conversely, a simple univariate regression model among those exposed (model 1 above) provides an unbiased estimate of the slope of the outcome versus the level of exposure . The estimated slope is also the same in a 2-factor model with an additional binary variable for exposed versus not and an intensity value zero for the nonexposed (model 3). The coefficient for exposure versus not in this model is the difference between the intercept of model 1 and the mean value of the outcome among the nonexposed (See Figure 1), and when the two are equal. Thus, in model 3 is not a very meaningful quantity.

Often the quantitative component is centered by subtracting the mean of among the exposed from the value for each exposed subject, while retaining the value zero for nonexposed subjects. The resulting coefficient for the centered intensity in the centered models 4 and 6 is the same as that for the noncentered intensity in the noncentered models 1 and 3, as are the standard error and P values. Further, the 2-df model F tests are the same in the noncentered model 3 and the centered model 6.

As described above, the coefficient for the intensity in the noncentered model 2 with all subjects (and value 0 for nonsmokers) is biased relative to the coefficient in models 1 and 3 (as well as for in models 4 and 6). However, the like centered model 5 provides an unbiased estimate of the coefficient for equal to that in these other models. Nevertheless, we also show that the estimated coefficients for other covariates added to this model might still be biased. Essentially, if there is an overall difference between the exposed versus nonexposed, such that the 2-factor model is correct, then fitting a model with only the centered exposure and other covariates is a misspecification with an important omitted covariate (exposed versus not) such that the estimated coefficients for the other covariates are then biased.

Further, the coefficient for exposure versus not in the centered 2-factor linear model 6 equals the mean difference of the outcome for those exposed versus nonexposed. This is a much more useful quantity than is the coefficient estimate in the noncentered model 3 (see Figures 1 and 2).

The same general results apply to a logistic regression model where the centered model coefficient for the binary component is the model-based log odds ratio of the outcome for those exposed versus nonexposed, and to a Cox proportional hazards model where is the overall log hazard ratio for smokers versus nonsmokers.

Thus, in general, a 2-factor model analysis with centered values for a semiquantitative variable (model 6 above) is recommended to assess the contributions of both the binary component (exposed versus not) and the quantitative component jointly, and to test the significance of the 2 components individually and jointly. Further, the coefficient for exposure versus not has the appealing property of comparing the outcome for those exposed versus not. This model will also provide unbiased estimates of the coefficients for other covariates added to the model.

Further, the 2-df F test above in model 6 (or model 3) provides a joint test of the total association of an exposure with the outcome. If that test is significant at level , then under the closed testing principle (12), additional tests of and can also be tested at the same level without the need for a correction for 2 tests. The same tests using the noncentered model 3 would be identical.

Robertson et al. (13) described 2-factor logistic models for a case-control study to capture a nonlinear association of the quantitative component with the outcome. Greenland and Poole (14) debated whether the nonexposed should be included and suggested a “practical compromise” that would fit the equivalent of models 2 and 3 above. However, we show that inclusion of the nonexposed with value zero in model 2 is ill-advised because it leads to a biased estimate of the slope.

A semiquantitative variable has also been called a variable with a “spike at zero.” Authors have described various approaches to model the association of the quantitative (nonzero) component with the outcome as nonlinear functions, such as fractional polynomials (15–17). Some have used the binary component to represent the nonexposed (=1) versus the exposed (=0). In terms of the notation herein this yields the centered 2-factor linear model

(14)

Compared to the centered model 6, and so that is the difference between the mean outcome among nonexposed minus exposed.

Recently Lorenz et al. (18) also compared models with linear trends versus nonlinear fractional polynomials and showed that the latter, in general, provide better fit than simple models equivalent to models 2 and 3 herein. However, their focus was on the deviance of the fit of different dose-response models, not the interpretation of the coefficients. Similar results would apply to other “model-free” methods to describe the association of the outcome with the quantitative exposure such as regression splines.

In the present work, we have assumed that the association between the outcome and the intensity among those exposed is captured by a linear risk gradient. In practice it would be prudent to also employ model-free methods to describe the pattern or nature of the risk gradient and to then employ a model with a risk gradient that best fits the observed data. In doing so, however, it is important to start with an unbiased model such as the simple model 1 restricted to those exposed, or the 2-factor model 6 (or 3). This is especially important when the analysis is attempting to establish the causal association of the exposure with the outcome.

In conclusion, we have described the properties of different simple models to assess the association of a semiquantitative exposure with an outcome, and describe the benefits of employing separate coefficients to represent the influence of the binary component for any exposure versus not and the influence of the quantitative level of exposure. Such a model provides an unbiased assessment of the overall effect of the exposure. Further, when the quantitative exposure values are centered, the coefficient for the binary component then equals the difference in the outcome between the exposed versus not, either in the means of a quantitative outcome, the logits of a probability (or log odds ratio) for a binary outcome in a logistic model, or the log hazard ratio for a Cox proportional hazards model of lifetimes. Further, in a centered 2-factor model, the coefficient estimates for other covariates are unbiased. Thus, the preferred approach is model 6 with a binary variable to represent exposure versus not and a centered quantitative covariate to represent the amount of exposure (with value zero for the nonexposed), and other covariates of interest.