Structural equation analysis of the causal relationship between health and perceived indoor environment

Abstract

Objectives

To explore the temporal relationship and reversed effects between health and perception of the indoor environment using structural equation models.

Methods

The study was a two-phase prospective questionnaire study with a cross-lagged design. Altogether 1,740 adults participated on both occasions.

Results

The perceived indoor environment had only weak effects on health at follow-up. However, the results strongly indicated a reversed effect that health problems may lead to increased complaints about the indoor environment.

Conclusions

Structural equation models are powerful analytical tools for disentangling the effects of a specific variable on another in high dimensional data with complex patterns of associations. The analyses confirmed the results of our previous logistic regression analysis about the strong reversed effect. Hence, it is probable that a reversed effect between health and complaints about the indoor environment exists.

Appendix: Technical details on the structural equation analysis

The model for mucous membrane symptoms is described in detail to illustrate the structural equation analysis. Thus, let Y _jkl denote the value at the jth examination (j = 1, 2) of the kth (k = 1, ..., 5) ordinal health variable for the lth subject (l = 1, ..., n). Similarly, let X _jml denote value on the mth (m = 1, ..., 9) indoor environment variable and let Z _1l , ..., Z _ql , denote values of q covariates measured at baseline. First, each ordinal health outcome Y _jkl was linked to a latent continuous variable Y _jkl ^* using a threshold model. Thus, the value of the ordinal variable is v (v = 1, ..., 4), if the underlying continuous variable falls in the vth interval, i.e., if τ _jk(v−1) ≤ y _jkl ^* ≤ τ _jkv , where the thresholds (τ) are unknown parameters to be estimated in the analysis (Muthen 1984). Then the continuous outcomes (Y _jkl ^*) were assumed to depend on common latent variables η _1l and η _2l representing the latent mucous membrane symptoms at baseline and follow-up respectively, i.e., $Y_{jkl}^{*}=\nu_{jk}+\lambda_{jk}\eta_{jl}+\epsilon_{jkl}.$ Thus, outcomes at baseline were all assumed to depend on η _1l , while outcomes at follow-up were all assumed to depend on η _2l (Fig. 1). In addition, all outcomes were assumed to be affected by a normally distributed random error ($\epsilon_{jkl}$). These error terms are often assumed to be independent (Bollen 1989), but here we allowed for correlation in error terms in the same variable at two different occasions, i.e., cov($\epsilon_{1kl},\;\epsilon_{2kl}$) ≠ 0.

The relationship between the mth indoor environment variable and the health outcomes illustrated in Fig. 2 were modeled using four equations:

$$ \eta _{{2l}} = \alpha _{4} + {\sum\limits_{g = 1}^q {\gamma _{{4g}} Z_{{gl}} + \beta _{1} X^{*}_{{1ml}} } } + \beta _{2} \eta _{{1l}} + \varsigma _{{4l}} $$

(1)

$$ X^{*}_{{2ml}} = \alpha _{3} + {\sum\limits_{g = 1}^q {\gamma _{{3g}} Z_{{gl}} + \beta _{3} X^{*}_{{1ml}} } } + \beta _{4} \eta _{{1l}} + \varsigma _{{3l}} $$

(2)

$$ \eta _{{1l}} = \alpha _{2} + {\sum\limits_{g = 1}^q {\gamma _{{2g}} Z_{{gl}} + } }\varsigma _{{2l}} $$

(3)

$$ X^{*}_{{1ml}} = \alpha _{1} + {\sum\limits_{g = 1}^q {\gamma _{{1g}} Z_{{gl}} + } }\varsigma _{{1l}} $$

(4)

where X _1ml ^* and X _2ml ^* are underlying continuous versions of the observed ordinal environment variables. Thus, perceived environment and mucous membrane symptoms at follow-up were assumed to depend linearly on the variables measured at baseline. The strength of these effects are reflected by the values of the parameters β ₁, ..., β ₄ each corresponding to a single headed arrow in Fig. 2. Standardized effects were obtained by multiplying β with the standard deviation in the predictor and deviding by the standard deviation in the outcome. The covariates were allowed to affect all four variables. Residual variation was modeled using ζ-variables, which were assumed to follow a normal distribution with mean zero. Residual variation in variables collected at the same time-point were allowed to be correlated as indicated by the double headed arrows in Fig. 2. In the joint analysis including all nine indoor environment indices, Eqs. (2) and (4) were repeated for each index. Furthermore, Eqs. (1) and (2) were modified so that they allowed for linear effects of each of the indoor environment variables measured at baseline.