The propensity score is the conditional probability of assignment to a particular treatment given a vector of observed baseline covariates. Therefore, the propensity score can be considered a balancing scoreb(x), which is a function of the observed covariatesx such that the conditional distribution ofxgivenb(x) is the same for both treated and control groups.[13].
Large Sample Theory
Rubin and Rosenbaum (1983) stated that the theory of both large and small samples shows that adjustment for scalar propensity scores is sufficient to remove bias due to all observed covariates between subjects assigned to the respective treatments. The propensity score is used in non-randomized experiments so that direct comparison can be made, also for systematic differences between treatment groups.
Small Sample Theory
There has been no agreement in applied research about which variables should be included in a propensity score model. However, some of the suggested sets of variables included the following: all measured baseline covariates (known as the full propensity model), all baseline covariates related to treatment assignment, all covariates related to the outcome, known as potential confounders, and all covariates that are associated with both treatment assignment and the outcome (known as true confounders). The authors suggested that unless there is a consensus that a variable is unrelated to the outcome or is not an appropriate covariate, it is advisable to include it in the propensity score model.
The authors adjusted for propensity score by including the propensity score as a covariate in a multivariate outcome model. The results of the simulation showed that the optimal model was the one that included the confounder and the outcome-only variable (PS model 4). It was found that the variance of the estimated treatment effect increased when an exposure-only variable (X3) was added to the model.
Principal Component Analysis
The results confirm those found by Rubin and Thomas (1996).[1] One of the disadvantages of the above methods is the necessity to have knowledge of the associations between covariates and outcome. Without knowing covariate-to-outcome relationships, misspecification of the propensity model is likely and thus produces potentially large biases of covariate estimation. The first principal component is the first variables in the newly transformed set and is the normalized linear combination of the original variables with maximum variance; the second principal component is the normalized linear combination with maximum variance from all linear combinations uncorrelated with the first principal component, and the subsequent principal components proceed in the same way.
Therefore, principal component analysis is concerned with explaining the variability in a vector variable by replacing it with a new variable with a smaller number of components with a large variance. The coefficients in the first principal component are the components of the normalized eigenvector corresponding to the largest latent root, and the variance of the first principal component is this largest root. [19] A common criticism of this method of data reduction is that it is not invariant to linear transformations of the variables. Therefore, when variables are not measured in the same units, it is recommended to extract the principal components from the correlation matrix instead of the covariance matrix. (Hotelling, 1933) Based on this recommendation, the principal components used for this analysis will be extracted from the correlation matrix as opposed to the covariance matrix.[19].
Penalized Maximum Likelihood
Therefore, conditioning on the propensity score provides unbiased estimates of the true treatment effect if the treatment effect measure is differences in means or proportions. However, misspecification of the propensity score model results in smaller biases than misspecification of the response model. The author concluded by stating that the propensity score's value lies primarily in protecting against model misspecification.
Similarity between treated and untreated subjects of baseline covariates indicates that propensity score matching produced a matched sample. In this case, the propensity score distribution in untreated subjects had a heavier left tail, the support of the propensity score distribution being similar in treated and untreated subjects. The propensity score subclassification assigned 50% of the sample to physical therapy, and 50% to surgery based on their propensity score.
Therefore, from the B=1000 replicates, there are 1000 associated p-values for the treatment effect for each propensity score method. The performance of the propenity score model was defined as the ability to correctly identify a significant or non-significant treatment effect.
Treatment and Outcome associated with the same baseline co- variatesvariates
Therefore, using principal components instead of main effects did not benefit the propensity score model in this case. Finally, the PS 4 method, a penalized maximum likelihood propensity score, yielded only eight significant treatment effects out of 1000 simulations. The results also showed that when treatment and outcome were related to the three main effects of (1) baseline age, (2) fat pad level, and (3) tear size, all four propensity score methods were able to correctly identify a significant treatment effect for nearly 100% simulations.
PS Method 3 performed as badly as PS Method 1 in that it identified 100% of the treatment effects as significant. Therefore, using principal components instead of main effects did not favor the propensity score model in this case. Finally, PS Method 4, propensity score with penalized maximum likelihood yielded only 12 significant treatment effects out of 1000 simulations. However, when the three clinical main effects were not the same as those associated with SPADI, the propensity score model performed poorly.
Treatment and Outcome with no baseline covariates in common
The above results are evidence to suggest that when the three clinical main effects included in the propensity score model were correctly specified, meaning that they were truly related to the SPADI, the propensity score model that included the correct set of variables performed best . . We expect a propensity score method that includes main effects that are also associated with SPADI (PS method 1) to have the fewest spurious treatment effects. We would also expect PS Method 2 to have a higher number of misidentified significant treatment effects compared to PS Method 2 due to the fact that PS Method 1 does not include SPADI-related main effects, but does include treatment-related ones.
The results showed that when treatment and outcome were associated with three separate and distinct main effects, PS Method 1 correctly yielded zero treatment effects for 100%. Results showed that when treatment was associated with (1) tall size, (2) bicep tendonitis, and (3) muscle atrophy, and when SPADI was associated with (1) age at baseline, (2) degree of fat deposits, and (3) tear size , PS method 1 yielded correct significant treatment effects for 997 out of 1000 simulations, and PS methods 2, 3, and 4 yielded correct significant results for 386, 892, and 118 out of 1000 simulations, respectively. Main effects associated with SPADI are a subset of main effects associated with treatment associated with treatment.
Main effects associated with SPADI are a subset of main effects associated with treatmentassociated with treatment
In the event that the treatment is associated with the three pre-specified main effects of (1) age at baseline, (2) degree of fat accumulation and (3) tear size, (4) age at baseline, (5) degree of fat accumulation, fat deposits and (6) tear size, and SPADI is associated with three different effects: tall size, muscle atrophy, and biceps tendonitis, which is a subset of the treatment-related effects. We applied the four methods of variable selection and the performance of the different propenity scores to obtain a correct main effect for the treatment. We expect that the propensity score method that includes main effects also associated with SPADI (PS method 2) will have the lowest number of spurious treatment effects. We would also expect PS Method 2 to have a greater number of incorrectly identified significant treatment effects compared to PS Method 1, due to the fact that PS Method 2 does not include the main effects associated with SPADI, but does include those associated with adhere to the treatment.
The results showed that the PS 2 method correctly yielded non-zero treatment effects for all 1000 replicate simulations. The results showed that when treatment was associated with (1) long size, (2) biceps tendonitis, and (3) muscle atrophy, and when SPADI was associated with (1) baseline age, (2) degree of fat deposits, and ( 3) tear size, the PS 1 method gave correct significant treatment effects for 997 out of 1000 simulations, and the PS 2, 3, and 4 methods gave correct significant results for 386, 892, and 78 out of 1000 simulations, respectively. The results of both null and non-null treatment effects, and in both scenarios of associations, we show that it is important to determine the correct main effects when the outcome variable is associated with a subset of treatment-related variables. propensity score model.
Main effects associated with treatment are a subset of main ef- fects associated with outcome
We expect that the propensity score that includes the same three main effects as those associated with SPADI (PS Method 2) will yield the lowest number of falsely significant treatment effects. We also expect that PS Method 4 will yield a low number of falsely significant treatment effects due to the contraction of treatment effects by the penalized maximum likelihood. In the event that treatment is associated with the three pre-specified main effects of (1) tall height, (2) muscle atrophy, and (3) biceps tendonitis, we expect the propensity score method that does not include these main effects to produce effects to the highest number of significant treatment effects when the treatment and propensity score on SPADI is reduced.
We expect that PS method 4 will yield the lowest number of significant treatment results due to the reduction in treatment effects. The results showed that PS method 1, which does not include the three main effects associated with SPADI, yielded the highest number of significant treatment effects. This confirms our above results suggesting that variables related to treatment and outcome are important in propensity score modeling.
Mixed main effects for both treatment and outcome
This confirms the hypothesis that variables related to the outcome are essential to develop a useful propensity score. In the case where we know the associations of baseline variables with treatment and outcome, the propensity score model that includes variables related to the treatment or outcome performed better than the other propensity score methods. To avoid misspecification of the main effects, the propensity score model with maximum likelihood performed as well as the propensity score model with properly specified main effects.
Therefore, the recommendation based on our simulation results is to use penalized maximum likelihood propensity score models to avoid yielding a higher number of misidentified treatment effects. What are the reasons why the principal component propensity score model (PS method 3) performed significantly worse compared to the other three models. And finally, future research should investigate how omitting quadratic terms might affect the performance of these four propensity score methods.
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