Two biomarkers are available in regard to a child's prenatal mercury exposure. After a logarithmic transformation, the relation between mercury concentrations in cord blood and maternal hair is approximately linear. Therefore, the following model for the distribution of the exposure variables appears appropriate
where the subject index i has been suppressed for simplicity in notation and the log base is 10. The latent variable η1 represents the true prenatal mercury exposure and is assumed to follow a normal distribution. The model further assumes, that except for measurement error, the two exposure indicators are given as a linear function of the true exposure. Here the measurement error is a sum of two different types of error: laboratory measurement imprecision and biological variation. The second error component arises because the mercury concentration in the fetal circulation is not constant over time but varies according to maternal mercury intake. It may also include individual differences in the distribution of mercury in the body.
The measurement error terms εB-Hg
are assumed to be normally distributed with means 0 and variances
, respectively. Furthermore, the blood and hair measurement errors are assumed independent. Methylmercury is thought to have a biological half-life of 45 days or slightly more [4
], so the concentration present in the cord blood reflects the exposure mainly during the last couple of months of gestation. If the active dose is some sort of a long-term average mercury concentration, then the assumption of independence between measurement errors in cord blood and in maternal hair may be appropriate, because digested mercury is deposited in the hair with a lag time of up to 6 weeks. This lag-time may ensure that the two biomarkers are not affected by the same random biological fluctuations on a temporal scale. In addition, concentrations of mercury in hair and in cord blood were determined by two different laboratories [22
] which means that analytical errors are unlikely to be correlated.
For identifiability the cord blood factor loading is fixed at one (λB-Hg = 1), thus the true mercury exposure has the same scale as the (log-transformed) cord blood concentration. The mean of η1 is identified by fixing the intercet νB-Hg at zero. However, even with these restrictions, there are too many free parameters and the exposure part of model is not identified. Additional information on the prenatal mercury exposure is available from the questionnaire data on maternal pilot whale meat consumption during pregnancy. The distribution of the ordered categorical variable Whale (5 categories: 0,1,2,3,≥ 4) is modeled introducing a latent continuous variable (Whale*) and assuming a threshold relation. In this example, the continuous latent variable could represent the weight of ingested whale meat.
Intake of pilot whale meat differs fundamentally from the measurements of mercury concentrations in hair and blood. While the latter two are determined (with a certain measurement error) by the true exposure (η1
), it may seem more natural to consider pilot whale meat intake as a determinant of a true exposure: an increase in maternal whale meat intake will increase the mercury exposure, not the other way around. Bollen [7
] describes such response variables as cause
indicators as opposed to the two biomarkers which enter the model as effect
indicators. From (2) it is seen that latent variables can only be affected by covariates and other latent variables. Thus, to incorporate this cause indicator in the current modeling framework formally it is necessary to introduce an additional latent variable η2
. This latent variable is identical to Whale
* and assumed to affect the latent mercury exposure. With three indicators of the latent variable η1
, the exposure part of the model is identified.
For the neuropsychological test scores, the optimal structural equation analysis would assume a single latent outcome variable. However, with tests that spanned from computerized assessment of motor speed to delayed recall of nouns, the scores considered clearly depend on different functional domains. The effect variables were therefore sorted into major nervous system functions, with one group consisting of motor functions, the other group encompassing cognitive function with a verbal component. Thus, it is assumed that the scores on the NES-tests (FT1
) are all indicators of an underlying motor function (η3
), while the scores on BNT, the CVLT and Digit Spans are all indicators of a latent verbally mediated function (η4
). Although this categorization may appear as a severe simplification of diverse outcomes that may depend on multiple functional domains, this analytical approach may be reasonable given the multifocal or diffuse effects of mercury neurotoxicity. To define the scales of the two latent neurobehavioral functions, the factor loadings of the responses FT1
are fixed at one. In agreement with previous analyses performed by Grandjean et al. [5
], the neuropsychological outcome variables are all modeled as continuous (conditionally) normally distributed variables. As a starting point, the elements of the measurement error vector (εi
) are assumed independent.
The true mercury exposure is hypothesized to affect the two latent outcome functions negatively after adjustment for effects of covariates. Thus, the structural part of the model is η = α + Bη + Γz + ζ with
where β31 and β41 denote the effect of mercury exposure on the motor function and the verbally mediated function, respectively. These parameters indicate the effect of prenatal mercury exposure corrected for measurement error, and they constitute the parameters of main interest in this analysis.
Maternal whale meat intake is assumed to affect the child's mercury exposure, but no direct effects of Whale* (= η2) on the cognitive functions are present in the model (β32 = β42 = 0). In other words, the true mercury exposure is considered an intermediate variable in the relation between maternal whale meat intake and the child's neurobehavioral function.
Potential confounders of the association between mercury exposure and child test performance are included in the model as covariates. By constraining the appropriate Γ-coefficients (2) to zero, it is assumed that computer acquaintance has no effect on the verbally mediated test scores. None of the exposure-confounder associations are ruled out in advance. Thus, the means of the latent variables η1 and Whale* are assumed to depend linearly on each of the confounders.
The first component of the disturbance term ζ = (ζ1, ..., ζ4)t models the conditional distribution of the true mercury exposure given the covariates and intake of whale meat. The second component describes the conditional distribution of Whale* given the covariates. The last two components give the conditional distribution of the two latent neurobehavioral functions given the covariates and the latent mercury exposure. These two components are not assumed to be independent as motor and verbal functioning are expected to be positively correlated given the values on covariates and true mercury exposure. Figure gives the path diagram illustrating the initial model for these data.
Figure 1 Path diagram for the association between indicators of mercury exposure and childhood neurobehavioral functions. The latent true mercury exposure is assumed to be affected by the covariates and maternal pilot whale meat intake. The two mercury biomarkers (more ...)
Correction for local dependence and item bias
Unfortunately, the proposed model does not fit the data adequately when compared to the unrestricted model (
< 0.0001). The correlation structure assumed for the neurobehavioral test scores is clearly too simple. The assumption that a child's test scores are independent given the latent level on the neurobehavioral functions is violated here, because the eleven outcomes originate from only five separate test protocols. Scores from the same test protocol are likely to show local dependence.
Local dependence is now modeled introducing three new latent variables (η5,η6,η7), which enter the model as random effects. In addition to the latent motor function, the finger tapping tests (FT1, FT2 and FT3) are all assumed to depend linearly on η5, which is normally distributed with zero mean and independent of all other variables. This random effect can be interpreted as indicating how good the child is at the common task, key tapping, corrected for the more general motor ability. In the same way η6 and η7 describe local dependence for the BNT-scores and the CVLT-scores, respectively. The path diagram in Figure illustrates how local dependence between indicators of neurobehavioral functions is incorporated in the structural equation model.
Figure 2 Path diagram showing how local dependence between neuropsychological test scores is taken into account. Test scores originating from the same test protocol are allowed to show excess correlation in relation to the degree explained by the underlying neurobehaviroral (more ...)
As expected, none of the three random effects could be ignored: in (naive) Wald tests random effect variances are highly significant with u
-statistics between 5.11 (CVLT) and 6.59 (FT). Furthermore, all random effect factor loadings are highly significant (data not shown). Incorporation of local dependence improves the model fit substantially, but the fit of the unrestricted model is still significantly better (
In this analysis, a consequence of the assumption of no item bias is that the covariates are assumed to affect indicators of the same latent cognitive function in the same way except for scale differences. For example, the ratio between mercury corrected regression coefficients of a given covariate on the first two finger tapping tests is equal to the ratio of the motor function factor loadings (λFT1,3/λFT2,3). Comparisons of regression coefficients obtained in naive multiple regressions for each indicator suggested that the assumption of no item bias is not satisfied for the study outcomes.
Here item bias is identified successively for the covariates. For a given covariate, item bias parameters are included for all indicators expect FT1 and CV LT1, which are chosen as the reference outcomes. Parameters that are insignificant in successive tests (backward elimination) are removed from the model and a new covariate is considered the same way. The covariates are analyzed in the order indicated by Table , starting with covariates a priori thought to be most important (i.e., the child's age and sex and maternal intelligence). To avoid identification of spurious effects using this multiple testing procedure, only parameters with a numeric u-statistic above 2.5 were considered significant.
Estimated effects of the covariates on the latent motor function, the latent verbal function and on biased indicators.
The extended model with six item bias parameters gives a very good fit (
= 0.26). Despite the strong improvement in model fit, the estimated mercury effects (Table ) changed only slightly as a result of correction for local dependence and item bias. Thus, for the motor function it is estimated that the effect of a tenfold increase in the true mercury exposure corresponded to a loss of about 1 point on the finger tapping test with preferred hand (FT1
). For the verbally mediated function the effect a similar exposure increase corresponded to a loss of about 1.6 points on the cued BNT-score (BNT2
). The latter effect is highly significant with a p
-value below 0.002 while the motor effect is on verge of statistical significance using the conventional level of 5%.
Estimates of the effect of a ten-fold increase in mercury exposure on two latent neurobehavioral functions obtained in different structural equation models.
it is not possible to obtain a mean and variance corrected test (8) of the overall hypothesis of the no mercury effect (β31
= 0). Differences between mean and variance corrected test statistics may not follow a χ2
-distribution. However, exploiting that the estimators (
) asymptotically follow a normal distribution, an ordinary Wald statistic can be calculated for this hypothesis. The correlation between the two mercury effect estimators was estimated at 0.098, which means that
= 13.33 with p
= 0.0013. Using the WLS fit statistic (7) the hypothesis of no mercury effect can be tested directly. This test yielded a
-value of 31.13 corresponding to a p
-value below 1/106
. Thus, the overall test was clearly more statistically significant using WLS inference. Simulation studies have shown that the WLSMV has better statistical properties than the WLS [14
]. Therefore, the test based on the WLSMV estimates probably yielded the most reliable result. This was confirmed in later analyses using maximum likelihood inference.
Table shows estimated factor loadings (λ) and measurement error variances (ω2
) for the two biomarkers of prenatal mercury exposure. These estimates show very little variation across the models considered, therefore only the estimates of the model including adjustments for local dependence and item bias are given. The quality of an indicator is not determined solely by the measurement error variance. When indicators have different factor loadings the measurement error variances are on different scales and cannot be directly compared. The indicator with the largest error variance might be the best indicator if it also has the largest factor loading. The measurement error standard deviation of the maternal hair concentration was therefore converted to the scale of the cord blood concentration after multiplication by the absolute value of the factor loading ratio (ωH-Hg
|). From the converted error variances (Table ), it is seen that the cord blood mercury gives the most precise reflection of the true exposure. This result is in agreement with a priori expectations and with the results of Grandjean et al. [5
] showing that in multiple regressions the cord blood concentration generally was a stronger predictor of childhood cognitive deficits than the maternal hair concentration. However, the error variance of the cord blood indicator, corresponds to a coefficient of variation of 28%. This result is approximately four times the documented analytical imprecision [22
Estimated factor loadings, measurement error variances and converted variances (see text) for measurements of mercury concentrations in cord blood an in maternal hair.
After local dependence has been taken into account, the variance of most indicators are assumed to come from three different sources of variation: variation explained by the latent neurobehavioral function, variation due to the random effect of the test subgroup, and indicator specific variation. For each indicator, Table shows how the total variance is distributed on these three variance components. Thus, the first column of the table gives the percentage of the total variation explained by the latent neurobehavioral function, i.e., the so-called reliability ratio [24
]. From these data it is seen that the neurobehavioral indicators generally are noisy with relatively low values between 10.4% to 66.0%. The two BNT scores measure the verbally mediated function with the greatest precision. For the CVLT-scores, reliability ratios decrease from learning to delayed recall and recognition. The Digit Span test measures the verbally mediated ability level of a given child with the same precision as short-delay recall on the CVLT-test. According to the model, the CVLT recognition test (CV LT4
) is a poor indicator of verbal ability. Another possibility is of course that this test does not measure the same brain function as the other CVLT-scores. This explanation may also be appropriate for the motor indicator HEC
which has a reliability ratio about half that of the finger tapping tests.
Estimated parameters in the measurement model of neurobehavioral test scores.
The last column of Table illustrates how the definition of the latent neurobehavioral functions has changed after taking local dependence into account. For each test score, the ratio (in percent) between the reliability ratios with and without correction for local dependence is given. If this ratio is above 100%, then the indicator at hand measures the latent function with greater precision as a result of the correction for local dependence. This is seen to be the case especially for HEC and DS, which is not surprising. Both scores are alone in their subgroup. More weight is placed on such variables in the definition of the latent variables when extra correlation between the other indicators of the same latent variable is taken into account. The relative changes in reliability ratio may seem dramatic, but it should be noted that inclusion of local dependence changed the estimated mercury effects only slightly. Furthermore, effects of covariates (data not shown) also changed very little as a result of the correction for local dependence.
Six significant item bias parameters were identified (Table ). Four of these parameters regard item bias caused by the child's sex, i.e., that the relation between test scores in boys and girls differed between tests reflecting the same neurobehavioral function. In the original approach, where FT1 was chosen as the (unbiased) reference outcome, none of the parameters describing item bias of the child's sex could be removed for the motor indicators. However, the ratio of the bias parameters of FT2 and FT3 corresponded closely to the ratio of motor factor loadings (λFT2,3/λFT3,3). Thus, if FT2 (and not FT1) was chosen as the unbiased estimator then the coefficient of FT3 was clearly insignificant. This more parsimonious representation was therefore preferred in the final analysis. For the two BNT scores, which are on approximately the same scale (λBNT1,4 = 0.993, λBNT2,4 = 1), item bias of almost the same size was identified for the child's sex. For these outcomes another way to introduce item bias is to let the mean of the random effect (η7) depend on the child's sex. In this way item bias is introduced on a test-subgroup level using only one parameter.
Table shows the estimated effects of the covariates on the two latent neurobehavioral functions as well as the direct covariate effects on the biased indicators. As before, all regression coefficients of motor responses are on the scale of the FT1-test, while all regression coefficients of verbal responses are on the scale of BNT2. At the time of examination the ages of the Faroese children spanned from 6.3 years to 8.2 years, and age is a strong predictor of a good test performance. The relation between achievement levels of boys and girls varied for the motor outcomes. The general trend (as expressed by FT2 and FT3) was that boys did better than girls (2.06 FT1 points). However, for FT1 the advantage of being a boy was significantly smaller (2.06 - 1.43 = 0.64 FT1 points), while girls had an advantage on the HEC error score. Variation in sex effects were also seen for the verbally mediated tests. Here girls generally performed slightly better than boys, but on DS the girls clearly got better results, while boys were better on the BNT. The mother's intelligence, i.e., her score on the Raven test, was a strong predictor of good verbal functioning, but predicted motor outcomes rather weakly, except for the scores on HEC. Presence of major medical risk factors for neurobehavioral dysfunction was negatively associated with neurobehavioral functioning. Children in day care had a strong advantage on the verbal tests, while a slight disadvantage was seen on the motor tests. Vocational or professional education of each parent and the employment status of the father were weakly associated with motor ability. Stronger positive effects of these variables were seen for the verbal outcomes, but only the effect of paternal education was significant at the 5% level. For CV LT2 (short-term recall), paternal employment status was a very strong predictor. For both latent neurobehavioral functions the child's residence at the time of the examination was on the verge of being significant, indicating that urban children did slightly better than rural children. Finally, as expected, a strong positive effect was seen of computer acquaintance on the performance on the computer assisted tests.
ML estimation, missing data analysis and PCB correction
The aim of the following analysis is to estimate the mercury effect after correction for the effects of prenatal exposure to PCB. Unfortunately, for about half of the children no biomarker information is available on the PCB exposure. In standard analysis only children with complete information on all variables (complete cases) are considered. This is not an optimal solution, because information about the mercury effect is needlessly lost when attention is restricted to children with a PCB value. In Mplus it is possible to conduct an analysis, which takes into account also the incomplete cases, and which yields consistent estimation under the weaker assumption that data are missing at random. Before the PCB variable is included, a missing data analysis is performed to investigate the appropriateness of the underlying assumption of the previous complete case analysis that data are missing completely at random.
Mplus only allows missing data analysis in models where all response variables can be considered to be continuous and normally distributed given the covariates. In the structural equation model developed, only the variable on the maternal whale meat intake is considered ordinal. After a transformation (t(x) = log(x + 1)) this variable is approximately linearly associated with the cord blood mercury concentrations (the best indicator of true exposure). A model where all response variables are continuous was then obtained by replacing the original ordinal variable by the transformed counterpart.
This multivariate normal model fitted the data adequately. The likelihood ratio test against the unrestricted model yielded a p
-value of around 1% and an RMSEA
(9) of 1.9% with an upper 90% confidence limit of 2.6%. Furthermore, parameter estimates changed only slightly as a result of replacing the ordinal variable and changing the estimation method from weighted least squares (WLSMV) to maximum likelihood (ML). Table shows ML estimates of the mercury effects on the two latent neurobehavioral functions. It is also noticed that the estimated standard deviations of the ML estimates were slightly higher than the standard deviations of WLSMV estimates. This finding may seem a little surprising because the WLSMV is expected to be less efficient. With the WLS method, estimated standard deviations (data not shown) were even lower than with WLSMV, thus again indicating that inference based on this method may be too optimistic. This observation is further supported by the overall test of no mercury effects. In the continuous model, the likelihood ratio test statistic was 13.61, which when evaluated in a
-distribution yielded a p
-value of 0.0011. This result is in good agreement with the overall test based on WLSMV statistics (
= 13.33 with p
= 0.0013), but clearly not as significant as the possibly exaggerated WLS result given above (
= 31.13 with p
As already mentioned, when the covariates have missing values, a model is needed for the distribution of covariates in addition to the structural equation model. The standard solution in Mplus is to assume that the covariates follow a multivariate normal distribution. However, this assumption is not appropriate in the current data where most covariates are dichotomous. For the variables considered so far (i.e., disregarding the PCB exposure), 706 of 917 children constitute complete cases. Of the incomplete cases, 71 children have missing covariate information. However, the variable on the maternal Raven score is clearly the largest source. If this variable is disregarded only 14 children have incomplete covariate information. To avoid unreasonable model assumptions these 14 children are excluded in the following analysis. Thus, the remaining 903 children have complete covariate information except for the maternal Raven score. However, an ordinary multiple regression analysis revealed that, given the other covariates, the scores on the Raven test with good approximation can be assumed to follow a normal distribution. To obtain a data set without missing values on the covariates, the maternal Raven score was therefore removed from the set of covariates to the set of response variables. This was done without changing the structure of the relations between the maternal Raven score and the other variables and under the assumption that this response variable was measured without error.
Table also gives the estimated parameters of the structural equation after including children with incomplete information. It is seen that these are not markedly different from the estimates of the complete case analysis, indicating that data are missing completely at random. The estimated adverse effect of mercury exposure on verbal functioning dropped to the level of the weighted least squares analysis, while the estimated effect on the motor function became slightly stronger. As expected, the estimated standard errors of the estimates decreased after taking (almost) all available information into account. As a consequence, both mercury effects reached statistical significance at the 5% level.
At this point the PCB exposure was included in the model in place of the mercury exposure. Because the PCB exposure indicator has missing values it cannot be included without making distributional assumptions. After a logarithmic transformation, complete case regression analysis indicated that the PCB exposures are approximately normally distributed (given the covariates) with a linear relation to the neuropsychological test scores. Thus, the PCB exposure entered the model as a response variable, assumed to be affected by the covariates as well as maternal intake of whale meat. Because the measurement error in the PCB variable is not taken into account here, the estimated PCB effects may be biased low (numerically), but the significance tests are likely to be valid.
From Table it is seen that the estimated PCB effect on the motor function was very weak if at all present. The PCB effect on the verbally mediated function was stronger and just significant at the 5%-level. This result is in good agreement with the results obtained using ordinary complete case multiple regression analysis without correcting for the mercury effect [20
]. For the neuropsychological tests considered here, this standard analysis showed significant (p
< 0.10) PCB effects only for the two BNT-scores (BNT1
Maximum likelihood estimates of the effect of a ten-fold increase in prenatal PCB exposure on two latent neurobehavioral functions.
The estimated effects of mercury and PCB may be compared using standardized coefficients. For the verbally mediated function, the standardized effect estimate of the PCB exposure was -0.10. Thus, if the PCB exposure is increased by one standard deviation then this cognitive ability is decreased by 0.10 standard deviations. For the mercury exposure the corresponding number was -0.14. The standardized effect on the motor function was -0.01 for PCB and -0.11 for mercury. It should be noted that only the mercury effects was corrected for measurement error.
Indicators of mercury and PCB were then included in the same structural equation model to allow estimation of the individual effect of both exposures. The two sets of indicators entered the model as in the separate analyses, taking into account that exposure to PCB and mercury may be correlated given the confounders and the variable on maternal whale meat intake. While the PCB exposure was first assumed to be measured without error, this assumption is clearly not realistic. Still the long half-life of PCB congeners as compared to that for methylmercury should lead to an exposure indicator less sensitive to short-term fluctuations in maternal marine food intake. However, normal analytical imprecision could easily be 10% (coefficient of variation), to which some biological variation would be added.
When estimating the mercury effect adjusted for possible effects of PCB exposure it is important to take the imprecision in the PCB marker into account. As a result of the strong correlation between exposure levels to mercury and PCB, failure to correct for PCB measurement error can lead to de-attenuated estimates of the mercury effect [6
]. Only one biomarker of PCB exposure is available, which means the total measurement error in this indicator cannot be identified in the structural equation analysis. Instead, the significance of PCB measurement error for inference on the mercury effect was investigated in sensitivity analyses assuming different values for the PCB measurement error variance (Table ). The marginal variances of the PCB concentrations (log transformed) and the cord blood mercury concentrations (log transformed) are approximately equal, so the two exposure indicators have about the same reliability ratio if the log10
(PCB) measurement error variance is assumed to be 0.02 (i.e. a coefficient of variation of
). If instead a log10
(PCB) measurement error variance of 0.04 is assumed, then the reliability ratio of the PCB exposure indicator is about the same as that of the maternal hair mercury concentrations. Figure shows the path diagram of the structural part of the model including exposures to both mercury and PCB.
Estimated effects of a ten-fold increase in exposure to mercury and PCB for different values of the PCB measurement error variance.
Figure 3 Path diagram illustrating how exposure to PCB is included in the analysis. After a logarithmic transformation the observed PCB concentration is assumed to give an error prone reflection of the child's true expsoure represented by the latent variable η (more ...)
Perhaps somewhat unexpectedly, it is seen from Table that the mercury regression coefficient on motor function was de-attenuated when adjusted for the effect of prenatal PCB exposure. This may indicate that the model was not strong enough to allow simultaneous analysis of these correlated exposures. On the other hand the mercury coefficient was still significant, which would typically not be the case in situations with multicollinearity problems. Residual confounding represents an alternative explanation of the de-attenuated mercury coefficient. When the size of the PCB measurement error was increased the estimated adverse mercury effect increased further, but at the same time it also became less significant.
For the verbally mediated function the mercury-corrected PCB effect was strongly attenuated and far from being statistically significant no matter how large the PCB measurement error was assumed to be. However, as expected, the PCB coefficient was negative, and the mercury effect was attenuated after the PCB correction. This attenuation became stronger the larger the PCB measurement error variance was assumed to be, and the mercury p-value was also sensitive to assumptions about the PCB measurement error. Thus, the mercury effect was significant (5% level) when the PCB indicator was assumed to be error free, but it became insignificant assuming that the error coefficient of variation in the PCB measurement was 46%. The same tendency was seen in the overall test for no mercury effects. In all analyses, the PCB effect remained far from significant.
Validation of the unrestricted model
So far, the models considered have been tested only against the unrestricted model, but the assumptions of this larger model should also be checked. The ordinal exposure indicator Whale was replaced by a continuous variable, with only minimal changes in the main results. Thus, the appropriateness of the unrestricted model where all response variables are continuous was therefore considered. Residual plots (not shown) indicated that, given the covariates the distributions of most responses were approximately normal. One indicator (CV LT4) deviated from normality with too many children achieving the maximum score. However, the main results did not change when this variable was excluded. Furthermore, the robustness of the inference on the mercury effect to the assumption of multivariate normality was investigated by calculating robust standard deviations (3) for the ML estimates. This approach yielded standard deviations of 0.571 and 0.514 for the mercury effect on the motor function and the verbally mediated function, respectively. These standard deviations are calculated in a complete case analysis, and should therefore be comparable to the standard deviations given in Table (ML estimation after full adjustment). The robust standard deviations are very similar to the ones obtained using normal distribution theory, indicating that the main result of this analysis is robust to the assumption of multivariate normality.
In addition to assumptions about multivariate normality of residuals, the unrestricted model assumes that the observed variables are linearly related. The appropriateness of the logarithmic dose response model for the effect of the two mercury biomarkers on the neurobehavioral outcomes has been carefully investigated using standard multiple regression methods (Budtz-Jørgensen et al., 1999, unpublished results). Likewise, regression analyses failed to identify significant differences of mercury effects in boys and girls [5
]. The strong effect of the child's age on the neurobehavioral test scores was investigated by including higher order terms. No important deviations from linearity were found.
The influence of mercury exposure on the definition of the neurobehavioral functions
In the models considered above, the parameters defining the latent variables are estimated simultaneously in joint analyses of all indicators. Using this approach the mercury exposure indicators may affect the measurement parameters of the two neurobehavioral functions. In other words the meaning of the latent constructs 'motor' and 'verbal' may depend on the exposure variables in addition to the neurobehavioral indicators. The risk that this influence is substantial may be reduced by the fact that all models considered are identifiable even if the exposure variables are disregarded.
The influence of the exposure indicators on the definition of latent neurobehavioral functions and vice versa may be investigated as follows. Two separate analyses were performed based on the multivariate normal model with adjustment for local dependence and item bias. First the parameters were estimated after exclusion of the exposure variables. Then the model was fitted again, this time disregarding the neurobehavioral indicators. In this way two sets of parameters were obtained in which the exposure indicators could not affect neurobehavioral parameters and vice versa. Finally, the model was fitted to all indicators fixing the factor loadings (Λ), the residual variances (Ω) and the parameter describing the effect of pilot whale intake on mercury exposure (β12) at the values obtained from the separate analyses. The variances of the latent variables incorporating local dependence were also fixed, but the residual variances of latent exposure and the latent neurobehavioral functions were kept free. Covariate effects were not fixed since their interpretation depends on whether they are corrected for the exposure effect. The result of the analysis with fixed parameters and the corresponding analysis without parameter constraints are given in Table . It is seen that the estimated mercury effects are only slightly attenuated in the fixed analysis, indicating that latent neurobehavioral functions are defined almost entirely by the neurobehavioral indicators, and that the latent exposure variable likewise is virtually unaffected by the outcome parameters.
Estimates of the effect of a ten-fold increase in mercury exposure.
As a final consideration, the results of the structural equation analysis not corrected for the PCB effect are compared to the results obtained using standard multiple regression analysis. Table shows estimated mercury effects obtained in complete case multiple regression analysis for the two main indicators of the exposure. These results differ from those previously published [5
] because the covariate Town7
has been added to the set of potential confounders. The cord blood coefficient of the indicators FT1
are on the same scale as β31
, respectively, of the structural equation models. It is seen that the two sets of parameters are approximately equal. Since the parameters of the structural equation model are corrected for measurement error in the exposure variables it may seem a little surprising that these coefficients are not numerically larger than the naive regression coefficients. This attenuation is caused by the introduction of the latent neurobehavioral functions that take into regard several test results. In a structural equation model with a latent exposure, but with no assumptions on the covariance matrix of the residuals of the neurobehavioral outcome variables, the estimated coefficients corresponded closely to the naive regression coefficients corrected for the estimated amount of measurement error in the exposure variables (data not shown).
For two biomarkers the effect of a ten-fold increase in prenatal mercury exposure on neurobehavioral outcomes is estimated in standard multiple regression analysis.
A serious weakness of the standard analysis is that the result is quite complex. Table contains 22 regression coefficients each on its own scale. Some coefficients are seen to be highly significant while others are clearly not. With 22 tests of the hypothesis of no mercury effect it is not surprising that some coefficients are significant. Thus, although the regression coefficients all suggest that the exposure is associated with a neurobehavioral deficit, it is not immediately clear form the standard analysis output whether or not the mercury effect is 'overall' statistically significant. For each of the exposure indicators, an overall test of the mercury effect may be obtained in a multivariate regression model assuming that the residuals of indicators are normally distributed with an unrestricted covariance matrix. The significance of the mercury effect is then assessed by testing the hypothesis that the mercury coefficient is zero for all outcome variables. This test was significant with a p-value of 2.45% for the cord blood indicator, while the test yielded a p-value of 9.70% for the maternal hair indicator. For comparison, in the structural equation analysis, the overall test was clearly significant with a p-value of 0.13%. Thus, in addition to providing a simpler presentation of the results, the structural equation approach yielded a stronger analysis.