Effect of variability on the estimation of relative hazards associated with other covariates

The effect of variability in the CD4 count on the relative hazard estimate is well documented and has been confirmed in this thesis. However, of interest is the effect that this variability has on the relative hazard estimates of other covariates included in the proportional hazards model. For example, if the relative hazard for the CD4 count is underestimated because of this variability, then the importance of other covariates in the model after adjustment may be overstated.

In order to assess these effects I have chosen to consider two covariates. Firstly, I will consider a fixed covariate at the time of seroconversion. I will call this a co-factor' effect although the covariate could represent any fixed covariate which has an effect on the CD4 count but no other effect on AIDS-free survival (e.g. a treatment). Secondly, I will consider the effect of a second time-dependent covariate which has no independent effect on AIDS-free survival, but which may be correlated with the CD4 count. Again, I will call this a ‘laboratory marker’, but it may represent any other time-updated covariate which is related to the CD4 count in some way.

For each of these analyses, CD4 counts are assumed to be measured monthly and the probability that they are missing is related to the underlying CD4 trend as described earlier. Results for k=1/4, 1 and 4 are included as these are sufficient to illustrate the pattern of the results.

7.6.1 A co-factor which Improves the immune status o f patients

I will firstly consider a co-factor which has the effect of raising the underlying CD4 count at baseline, in this case at seroconversion, but which doesn't have any effect on the subsequent rate of CD4 decline. I will then consider a co-factor which slows down the rate of CD4 decline but which doesn't change the absolute level at seroconversion. In both cases any effect of the co-factor on AIDS-free survival acts solely through the CD4 count. Once the CD4 count is adjusted for in the proportional hazards model, therefore, no residual effect of the co-factor should remain and its relative hazard should be approximately equal to one.

For the first scenario, the co-factor is assumed to raise the underlying CD4 trend at seroconversion by a mean of six units on the square root scale. Individual responses to the co-factor are assumed to vary around this normally with a standard deviation of three units. Hence, for a patient initially with a CD4 count of 500 cells/mm^, the co-factor raises the underlying CD4 count at seroconversion to 804 cells/mm^ on average whereas for a patient with only 200 cells/mm^ at seroconversion the therapy raises the CD4 count to only 406 cells/mm^ on average. The size of this effect is chosen so that the unadjusted co-factor relative hazard is large and that the effects of adjustment on the relative hazard can easily be seen.

Unadjusted for the CD4 count (top half of Table 7.4) this co-factor appears to lead to a 31% reduction in the hazard of AIDS. After adjusting for the CD4 count, either with mild (k=1/4) or moderate (k=1) variability, an increase in the relative hazard estimate is seen.

interest is the finding that the median relative hazard estimate is now greater than one, suggesting a detrimental effect of the co-factor after adjustment for the CD4 count. When heavy variability (k=4) is added to the CD4 count, the results remain heavily biased with both the adjusted relative hazard estimate and the upper limit of the 90% range remaining below one.

Table 7.4 : The effect of variability in the measurement of the CD4 count on the relative hazard estimate of a co-factor which improves the immune status of patients by (i) raising the CD4 count at seroconversion and (ii) slowing the rate of CD4 decline. CD4 counts are measured monthly and the proportion missing depends on the underlying CD4 trend

Effect of co-factor Variance Median 90% range

is estimate of of hazards multiplied relative

by ; hazard Raise CD4 count | Unadjusted

at seroconversion 'co-factor effect

0.69 0.54 - 0.86 , Adjusted co-factor 1 effect 1 1 1/4 1.20 0.98- 1.51 1 1.11 0.96-1.33 4 0.81 0.68-0.99

Slow rate of CD4 | Unadjusted

decline 'co-factor effect

0.68 0.53 * 0.84 1 Adjusted 1 co-factor effect 1 1 1/4 0.92 0.75- 1.11 1 0.87 0.72- 1.01 4 0.73 0.61 -0.88

For the second scenario, the co-factor is assumed to slow the rate of VCD4 decline by, on average, 0.5 units per year. Individual responses vary around this normally with standard deviation 0.25 units. Unadjusted for the CD4 count, this co-factor again appears to reduce the hazard of AIDS considerably. After adjustment for the CD4 count, with either light or moderate variability added, the relative hazard estimates still overstate the residual co-factor effect. However the 90% ranges include one suggesting that in many cases the correct conclusion regarding the effect of the co-factor would be reached. After adjustment for the CD4 count with heavy variability, however, the relative hazard estimate still suggests a large residual co-factor effect and the upper limit of the 90% range remains below one.

7.6.2 A second laboratory marker

Very often the prognostic values of other laboratory markers, which may be correlated with the CD4 count, are of interest. These markers are often measured at the same timepoints as the CD4 count. For this analysis I will assume that a second laboratory marker is measured perfectly and values will be missing whenever the CD4 count is missing. The second marker in itself is assumed to have no independent effect on AIDS-free survival. However, any correlation between this marker and the underlying CD4 trend will result in apparent prognostic value for the development of AIDS in analyses where the CD4 count is not adjusted for. After adjustment for the CD4 count, however, no residual effect on AIDS-free survival should remain.

The top section of Table 7.5 shows the results for a laboratory marker which is

uncorrelated with the CD4 count. Adjusting for the CD4 count would not be expected to have any great effect on the estimate of its relative hazard. Unadjusted for the CD4 count this laboratory marker has no prognostic value for the development of AIDS. After adjustment there are no consistent patterns to the relative hazard estimate and any differences are likely to be due to random variation rather than any other effect. The range of estimates is wide, however, both before and after adjusting for the CD4 count, which is again most likely a function of the variability introduced when generating 200 different data sets for each analysis. I will address this issue in the Discussion. The middle section of Table 7.5 shows the results for the situation where the second laboratory marker is moderately correlated with the CD4 count (correlation coefficient of between 0.3 and 0.4 at each time point). For this and the following analysis, the marker was generated by adding a random amount to the underlying CD4 trend; the larger this amount the smaller the correlation between the two markers. In a univariate analysis, it appears that a one unit drop in the second laboratory marker is associated with an 8% increase in the hazard of developing AIDS. Adjustment for the measured CD4 count, either when light or moderate variability is added, brings the relative hazard estimate down towards one. However, with heavy variability (k=4) the relative hazard estimate and the lower limit of the 90% range both remain above one, suggesting that there is a residual marker effect.

When the two laboratory markers are more highly correlated (correlation coefficients of 0.7 to 0.8), the results are more heavily biased (bottom section of Table 7.5). Unadjusted for the CD4 count, a one unit drop in the laboratory marker appears to be associated with a 42% increase in the hazard of developing AIDS.

When the CD4 count is measured with very little variability, the adjusted relative hazard correctly falls towards one. After adjustment for the CD4 count with either moderate or heavy variability, however, the relative hazards and lower limits of the 90% range remain above one, suggesting that there is at least an 7-32% increase in the hazard of AIDS per unit increase in the marker, even after adjustment for the CD4 count.

Table 7.5 : The effect of variability in the measurement of the CD4 count on the relative hazard estimate of a second laboratory marker which is (i) uncorrelated with the CD4 count, (ii) moderately correlated with the CD4 count and (iii) highly correlated with the CD4 count, but which does not in itself have any independent effect on AIDS-free survival. The CD4 count and the second laboratory marker are measured monthly and the proportion of both markers which are missing depends on the underlying CD4 trend

Correlation with CD4 count Variation is multiplied by: Median estimate of relative hazard 90% range for estimates None I Unadjusted 105 0.37 « 3.09 ["Adjusted for I CD4 count 1/4 1.02 0.43 - 2.65 1 1 0.94 0.39 - 2.58 1 1 4 1.00 0.42 - 2.54 Moderate 1 Unadjusted - 1,08 1.06*1.10 [^Adjusted for 1 CD4 count 1 1 1 1/4 1 4 1.00 1.02 1.06 0.99-1.02 1.00-1.03 1.05-1.08 High 1 Unadjusted - 1.42 1.37-1.48 ["Adjusted for 1 CD4 count 1/4 1.01 0.97-1.06 1 1 1.11 1.07-1.18 1 1 4 1.37 1 3 2 -1 .4 3