Power curve

Description

Determine the power curve between true and false models

Usage

mxPowerSearch(trueModel, falseModel, n=NULL, sig.level=0.05, ...,
                    probes=300L, previousRun=NULL,
                    gdFun=mxGenerateData,
                    method=c('empirical', 'ncp'),
                    grid=NULL,
                    statistic=c('LRT','AIC','BIC'),
		    OK=mxOption(trueModel, "Status OK"),
		checkHess=FALSE,
		silent=!interactive())

mxPower(trueModel, falseModel, n=NULL, sig.level=0.05, power=0.8, ...,
                    probes=300L, gdFun=mxGenerateData,
                    method=c('empirical', 'ncp'),
                    statistic=c('LRT','AIC','BIC'),
                    OK=mxOption(trueModel, "Status OK"), checkHess=FALSE,
                    silent=!interactive())

Arguments

Details

Power is the chance of obtaining a significant difference when there is a significant difference (1 - false negative rate). The likelihood ratio test is used by default. There are two methods available to produce a power curve. The default, method=empirical, works for any model where the likelihood ratio test works. For example, definition variables and missing data are fine, but parameters estimated at upper or lower bounds will cause problems. The method=empirical can require a lot of time because the models need to be fit 100s of times. An alternate approach, method=ncp, is much more efficient and takes advantage of the fact that the non-null distribution of likelihood ratio test statistic is often χ^2(df_1 - df_0, N λ), chi^2(df1 - df0, N lambda). That is, the non-centrality parameter, λ,lambda, can be assumed, on average, to contribute equally per row. This permits essentially instant power curves without the burden of tedious simulation. However, definition variables, missing data, or unconventional models (e.g., mixture distributions) can violate this assumption. Therefore, we recommend verifying that the output from method=empirical roughly matches method=ncp on the model of interest before relying on method=ncp. Unlike method=empirical, method=ncp does not use the gdFun argument and works with models that only have summary statistics for data.

When method=ncp, parameters of both ‘trueModel’ and ‘falseModel’ are assumed to be converged to their desired values. In contrast, when method=empirical, ‘trueModel’ need not be run or even contain data. On each replication, data are generated from ‘trueModel’ at the given parameter vector. Both ‘trueModel’ and ‘falseModel’ are fit against these data.

When statistic='LRT' then the models must be nested and sig.level is used to determine whether the test is rejected. For ‘AIC’ and ‘BIC’, the test is regarded as rejected when the ‘trueModel’ obtains a lower score than the ‘falseModel’. In contrast to statistic='LRT', there is no nesting requirement. For example, ‘AIC’ can be used to compare a ‘trueModel’ against its corresponding saturated model.

mxPower operates in many modes. When power is passed as NULL then power is calculated and returned. When power (as a scalar or vector) is given then sample or effect size is (are) returned. If you pass a list of models for ‘falseModel’, each model will be checked in turn and the results returned as a vector. If you pass a vector of sample sizes, each sample size will be checked in turn and the results returns as a vector.

In constrast to mxPower, mxPowerSearch attempts to model the whole relationship between sample size or effect size and power. A naive Monte Carlo estimate of power examines a single candidate sample size at a time. To obtain the whole curve, and simultaneously, to reduce the number of simulation probes, mxPowerSearch employs a binomial family generalized linear model with a logit link to predict the power curve across the sample sizes of interest (similar to Schoemann et al, 2014).

The mxPowerSearch algorithm works in 3 stages. Without loss of generality, our description will assume a sample size to power relationship, but a similar process is used for the parameter value to power relationship. In the first stage, a crude binary search is used to find the range reasonable values for N. This stage is complete once we have at least two rejections and at least two non-rejections. At this point, the binomial intercept and slope model is fit to these data. If the p-value for the slope is less than 0.25 then we jump to stage 3. Otherwise, we fit an intercept only binomial model. Our goal is to nail down the intercept (where power=0.5) because this is the easiest point to find and is a necessary prerequisite to estimate the variance (a.k.a. slope). Therefore, we probe at the median of previous probes stepping by 10% in the direction of the model's predicted intercept. Eventually, after enough probes, we reach stage 3 where the p-value for the slope is less than 0.25. At stage 3, our goal is to nail down the interesting part of the power curve. Therefore, we cycle serially through probes at 0, 1, and 2 logits from the intercept. This process is continued for the permitted number of probes. Occasionally, the p-value for the slope in the stage 3 model grows larger than 0.25. In this case, we switch back to stage 2 (intercept only) until the stage 3 model start working again. There is no other convergence criteria. Accuracy continues to improves until the probe limit is reached. After mxPowerSearch returns, additional probes can be run by passing the previous result back into the function as the previousRun argument.

When ‘n’ is fixed then mxPowerSearch helps answer the question, “how small of a true effect is likely to be detected at a particular sample size?” Only one parameter can be considered at a time. The way the simulation works is that a candidate value for the parameter of interest is loaded into the trueModel, data are generated, then both the true and false model are fit to the data to obtain the difference in fit. The candidate parameter is initially set to halfway between the trueModel and falseModel. The power curve will reflect the smallest distance, relative to the false model, required to have a good chance to reject the false model according to the chosen statistic.

Note that the default grid is chosen to exhibit the interesting part of the power curve (from 0.25 to 0.97). Especially for method=ncp, this curve is practically identical for any pair of models (but located at a different range of sample sizes). You should select your own grid points if you wish to align power curves from more than one power analysis.

Value

mxPower returns a vector of sample sizes, powers, or effect sizes.

mxPowerSearch returns a data.frame with one row for each ‘grid’ point. The first column is either the sample size ‘N’ or the parameter label. The second column is the power. The next two columns provide a +/-2 SE confidence interval for the power as estimated by the binomial logit linear model. When method=empirical then the ‘probes’ attribute contains a data.frame record of the activity of the power estimation process.

References

Schoemann, A. M., Miller, P., Pornprasertmanit, S. & Wu, W. (2014). Using Monte Carlo simulations to determine power and sample size for planned missing designs. International Journal of Behavioral Development, 38(5), 471-479.

See Also

mxCompare, mxRefModels

Examples

library(OpenMx)

data(demoOneFactor)
manifests <- names(demoOneFactor)
latents <- c("G")
factorModel <- mxModel("One Factor",
                       type="RAM",
                       manifestVars = manifests,
                       latentVars = latents,
                       mxPath(from=latents, to=manifests, values=0.8),
                       mxPath(from=manifests, arrows=2,values=1),
                       mxPath(from=latents, arrows=2,
                              free=FALSE, values=1.0),
                       mxPath(from="one", to=manifests),
                       mxData(demoOneFactor, type="raw"))
factorModelFit <- mxRun(factorModel)

indModel <- factorModelFit
indModel$A$values['x1','G'] <- 0.3
indModel$A$free['x1','G'] <- FALSE
indModel <- mxRun(indModel)

mxPowerSearch(factorModelFit, indModel, method='ncp')

mxPower(factorModelFit, indModel, method='ncp')