R: Power curve

mxPowerSearch {OpenMx} | R Documentation

## Power curve

### Description

`mxPower` is used to evaluate the power to distinguish between a hypothesized effect (`trueModel`) and a model where this effect is set to the value of the null hypothesis (`falseModel`).

`mxPowerSearch` evaluates power across a range of sample sizes or effect sizes, choosing intelligent values of, for example, sample size to explore.

Both functions are flexible, with multiple options to control n, `sig.level`, `method`, and other factors. Several parameters can take a vector as input. These options are described in detail below.

Evaluation can either use the non-centrality parameter (which can be very quick) or conduct an empirical evaluation.

note: During longer evaluations, the functions printout helpful progress consisting of a note about what task is being conducted, e.g. "Search n:power relationship for 'A11'" and, beneath that, an update on the run, the model being evaluated, and the current candidate value being considered, e.g. "R 15 1p N 79"

### 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

`trueModel` | The true generating model for the data.
`falseModel` | Model representing the null hypothesis that we wish to reject.
`n` | Total rows summing across all submodels proportioned as given in the trueModel. Default = NULL. If provided, result (power or alpha) solved-for at the given total sample size.
`sig.level` | A single value for the p-value (aka false positive or type-1 error rate). Default = .05.
`power` | One or values for power (a.k.a. 1 - type 2 error) to evaluate. Default = .80
`...` | Not used.
`probes` | The number of probes to use when method = ‘empirical’.
`previousRun` | Output from a prior run of ‘mxPowerSearch’ to build on.
`gdFun` | The function invoked to generate new data for each Monte Carlo probe. Default = `mxGenerateData`
`method` | To estimate power: ‘empirical’ (Monte Carlo method) or ‘ncp’ (average non-centrality method).
`grid` | A vector of locations at which to evaluate the power. If not provided, a reasonable default will be chosen.
`statistic` | Which test to use to compare models (Default = 'LRT').
`OK` | The set of status codes that are considered successful.
`checkHess` | Whether to approximate the Hessian in each replication.
`silent` | Whether to show a progress indicator.

### Details

Power is the chance of obtaining a significant difference when there is a true difference, i.e., (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(df1 - df0, N λ). 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`.

note: Unlike `method = empirical`, `method = ncp` does not use the `gdFun` argument and can be used with models that have summary statistics rather than raw 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. Then 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.

mxPowerSearch

In contrast 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 use the sample size to power relationship, but a similar process is used when evaluating the relationship of parameter value to power. 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 are no other convergence criteria. Accuracy continues to improves until the probe limit is reached.

note: After `mxPowerSearch` returns, you might find you wanted to run additional probes (i.e., bounds are wider than you'd like). You can run more probes without starting from scratch by passing the previous result back into the function using 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 show 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). However, if you wish to align power curves from more than one power analysis, you should select your own `grid` points (perhaps pass in the power array from the first to subsequent using `grid =` ).

Note: CI is not given for method=ncp: The estimates are exact (to the precision of the true and null/false model solutions provided by the user).

### 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’ (given as the proportion of rows provided in `trueModel`) or the parameter label. The second column is the power. If `method=empirical`, `lower` and `upper` provide a +/-2 SE confidence interval (CI95) 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 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, 471-479.

### See Also

`mxCompare`, `mxRefModels`

### Examples

    library(OpenMx)

    # Make a 1-factor model of 5 correlated variables
    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),
       mxPath(from= "one", to= manifests),
       mxData(demoOneFactor, type= "raw")
    )
    factorModelFit <- mxRun(factorModel)

    # The loading of x1 on G is estimated ~ 0.39
    # Let's test fixing it to .3
    indModel <- factorModelFit
    indModel$A$values['x1','G'] <- 0.3
    indModel$A$free['x1','G'] <- FALSE
    indModel <- mxRun(indModel)

    # What power do we have at different sample sizes
    # to detect that the G to x1 factor loading is
    # really 0.3 instead of 0.39?
    mxPowerSearch(factorModelFit, indModel, method='ncp')

    # If we want to conduct a study with 80% power to
    # find that  the G to x1 factor loading is
    # really 0.3 instead of 0.39, what sample size
    # should we use?
    mxPower(factorModelFit, indModel, method='ncp')

[Package OpenMx version 2.17.3 Index]