Original link:http://tecdat.cn/?p=23085
Introduction to the
In this paper, the efficiency of structural equation model is analyzed.
This article starts with a few simple examples. The rest of the section provides some statistical background, definitions of the various effect sizes, and detailed descriptions of the functions included in this package.
background
Statistical evaluation of mathematical models is usually performed by considering test statistics, which represent the difference between observed data and the data of the fitted model. In SEM, the relevant test statistics of samples of size N are given by T = Fˆ(n-1). Fˆ represents the minimum sample value of the selected difference function (such as maximum likelihood), thus indicating the mismatch between the model and the sample data. Therefore, T allows the likelihood ratio test for the null hypothesis (H0) that is correct for the model. If the hypothesized model holds in the population, it can be proved that T asymptotically follows the central χ2 (df) distribution, df=.5-p (p+1) -q degree of freedom, where p is the number of expression variables and q is the number of free parameters. This is why T is often referred to as the “chi-square model test statistic”.
Given the observation of a chi-square test statistic, a significance test for an invalid hypothesis can be performed. The usual inspection procedure is as follows. Given a specific α-error level (usually α=0.05), a critical chi-square value is obtained from the asymptotic central χ2 (DF) distribution. If the observed value of the chi-square test statistic exceeds the critical value, the null hypothesis that the model conforms to the data is rejected. Otherwise, H0 is preserved. Finding the observational statistics above the threshold (meaning that the upper tail probability is below the specified alpha level) leads to the statistical judgment that the difference between the hypothetical and actual population covariance matrix is too large to be attributable solely to sampling error. Thus, statistically significant chi-square test statistics provide evidence against the validity of the hypothetical model.
When using this framework to test statistical assumptions, two types of errors occur. The α error (or Type I error) for incorrectly rejecting a true H0 (a correct model) and the β error (or Type II error) for incorrectly retaining a false H0 (a wrong model).
If H0 is false, then the chi-square test statistic is no longer a central χ2 (DF) distribution, but can be proved to follow a noncentral χ2 (DF, λ) distribution with noncentral parameters λ and expected value DF + λ (MacCallum, Browne, & Sugawara, 1996). The noncentral parameter λ moves the expected value of the noncentral χ2 (df, λ) distribution to the right of the corresponding central distribution. After determining the critical value related to the expected alpha probability of the central χ2 (DF) distribution, a corresponding noncentral χ2 (DF, λ) distribution with a certain noncentral parameter λ can be constructed and the rightwing area of the distribution (i.e. Accordingly, the statistical efficiency is the area (i.e., integral) of the noncentral χ2 (df, λ) distribution to the right of the critical value. The general situation is shown in Figure 1.
Figure 1 depicts the central (solid line) χ2 (DF) and non-central (dashed line) χ2 (DF, λ) distributions, DF =200, λ=100. The area of the center distribution χ 2 (DF) to the right of the critical value reflects the α error. The solid line indicates a critical value of 234, corresponding to α=0.05. The dotted line indicates a critical value of 244 derived from the tradeoff efficiency analysis, which is associated with α=0.018. The χ2 (df, λ) distribution region to the left of the critical value is the β-error probability (β=0.006 and β=0.018 for the critical values 234 and 244). Statistical efficiency is defined as 1-β, that is, the area under the noncentral χ2 (df, λ) distribution to the right of the critical value.
Measure of the
To define the difference between the H0 and H1 models, any effect measure based on non-centrality can be used.
F0
F0 is the total minimum value of the maximum likelihood fitting function, which is defined as
Where Σ is the total covariance matrix of P, where is the implicit covariance matrix of P model, and P is the number of observed variables. If the model is correct, Σˆ= null, F0=0. Otherwise, F0>0, the larger the value, the greater the difference between the model and the data (mismatch).
RMSEA
Approximate root mean square error (RMSEA; Browne & Cudeck, 1992; Steiger & Lind, 1980) measures F0 by the degree of freedom of the model.
RMSEA is bounded by zero, with a lower value indicating a better fit. Implicit F0 is:
Defining an effect in RMSEA requires specifying degrees of freedom.
Mc
MC (McDonald, 1989) is the transformation of F0 on the interval from 0 to 1. The larger the value, the more suitable it is.
GFI
Fitting index (GFI; Joreskog & Sorbom, 1984; Steiger, 1990) scale F0 in the interval of 0-1, and the higher the value, the better the fitting degree.
Since GFI depends on the number of variables under observation (P), you need to provide this number when defining performance with GFI.
AGFI
The adjusted fitting index (AGFI; Joreskog & Sorbom, 1984; Steiger, 1990) modified the GFI to include a penalty for the number of free parameters, as measured by the degree of freedom of the model.
Using AGFI to illustrate the effect requires both the number of observed variables (P) and the degree of freedom (DF) of the model.
Methods that are not based on non-centrality
The fitting index which is not based on non-centrality has no direct relationship with F0, so it is not suitable for performance analysis. However, if the effect is defined by the H0 and H1 covariance matrices, at least the measurement can be calculated.
SRMR
The standardized root-mean square residual (SRMR) is a measure of the mean (square) difference between the (standardized) model and the total body covariance matrix, so it ranges from 0 to 1, with a smaller value indicating a higher degree of fit. Let E0 be the difference between the implied model and the covariance matrix of the population, E0=Σ- “, Vech represents the vecturation transformation, Q is the diagonal matrix with the dimension of.5p(p+1), including the inverse value of the product of the standard deviations of observation variables I and j, then SRMR can be defined as
The relationship between the residual matrices E0 and F0 is complex and depends on the implicit covariance matrix of the model, so SRMR is not well suited to define effects in terms of F0 (based on ML estimates).
CFI
The comparison fit index (CFI) is an incremental index that represents a reduction in the proportion of misfits of the hypothetical model (F0H) relative to the empty model (F0N), defined as a model that constrains all covariances to zero. In the aggregate, CFI ranges from 0 to 1, with higher values indicating better fit.
Although obtaining F0 from CFI is simple, this requires knowing F0N, which is difficult to determine a priori.
Power Analysis of performance Analysis
When performing a performance analysis, it is generally necessary to state the measure and size of the effect to be detected and to provide the model with freedom. Depending on the type of performance analysis, further evidence is needed. This section assumes that effects are specified in one of the above ways, and how effects are defined with the implicit and total body covariance matrix of the model.
A-priori: Determine the required N, given α, β, effect, and DF
The purpose of a prior efficacy analysis is to determine the sample size required to detect an effect with a given alpha error. In the language of structural equation modeling, a priori capability analysis asks, “How many observations do I need to have an effect? If my model is actually wrong (within the range defined by the selected effects), how many observations do I need to falsify my model with X% probability (force)?
Performing a priori efficiency analysis requires specifying the alpha error, the required efficiency (or, equivalent to the acceptable beta error), the type and size of the effect, and the model DF. Depending on the effect size metric you choose, you may also need to define the number of observed variables.
Assume that we want to detect the sample size required for a false statement of a model (involving df=100 degrees of freedom) with an efficiency of 80% with an α error of 0.05, where the loss of fit corresponds to RMSEA=0.05. Store the results in a list called ap1.
effect = .05, effect.measure = 'RMSEA',
alpha = .05, power = .80, df = 100The summary method is called on AP1 to output the result and the associated central and non-central chi-square distribution
This shows that N=164 produces about 80% efficiency to detect the specified effect. The output further shows the critical Chi-square, the non-central parameter (NCP), and the ratio between error probabilities (implied Alpha/Beta ratio). In this example, the ratio between Alpha and Beta is 0.25, indicating that Beta errors are four times more likely to be made than Alpha errors. This is clearly the result of the selected input parameter, since an efficiency of 0.80 (1-β) implies a β error of 0.20, which is four times the alpha error of the selected 0.05.
effect = .05, effect.measure = 'RMSEA',
alpha = .05, power = .80
Now we also obtained the equivalent value of GFI and AGFI of RMSEA=0.05, assuming df=100, p=20. It is also possible to specify an acceptable beta error instead of the desired performance. For example, call
effect.measure = 'RMSEA',
alpha = .05, beta = .20, df = 100, p = 20
The same output as above is given. If you are interested in efficiency changes over a range of sample sizes, it is useful to ask for efficiency diagrams, as detailed below.
Post-hoc analysis: Given alpha, N, effects, and df, determine the effectiveness achieved
The purpose of post hoc performance analysis is to determine the actual performance of a given sample size in detecting a particular effect under a certain α error. In the language of structural equation modeling, ex post efficacy asks, “How big is my sample? In the sample I have, if my model is indeed wrong (at least to the extent defined by the selected effects), what is the probability (power) of falsifying my model? A post hoc efficacy analysis requires the identification of alpha error, sample size, type and size of effects, and model DF. Also, depending on the effect size metric you choose, you may need to define the number of variables to observe. Assume that the performance we expect to achieve with a sample size of N=1000 is capable of detecting errors in the model (involving df=100 degrees of freedom), and the α error is 0.05, where the error fitting quantity corresponds to RMSEA=0.05. We store the results in a list called ph1.
posthoc(effect = .05, effect.measure = 'RMSEA',
alpha = .05, N = 1000
Calling the summary method on ph1 shows very high performance (performance >.9999). The associated error probabilities are provided with higher precision. Specifically, the beta error is beta =2.903302e-17, which translates to 2.9-10-17=0.000000000000000029. In practice, a model with RMSEA>=.05 (or F0>=0.25 or MC <=.882) works well under these conditions. The implied α/β ratio is 1.722177E +15, suggesting that an α error is 2 trillion (1015) times more likely than a β error. It is useful to ask for a performance graph if you are interested in a range of effects with varying degrees (for example, from 0.01 to 0.15 for RMSEA), as detailed below.
Compromise-Effectiveness Analysis: Given the α/β ratio, N, effect, and DF, determine α and β
The purpose of tradeoff efficiency analysis is to determine the threshold values of α and β (and the associated chi-square test statistics) for a given effect, a certain sample size, and the expected α and β ratios (Moshagen & Erdfelder, 2016). In the language of structural equation modeling, compromise analysis asks, “What do I do with the sample I have? In my sample at hand, how should I choose the threshold value for the chi-square model test to determine the corresponding α and β errors when my model is consistent with the hypothesis of full fit?
Suppose we want to determine the critical chi-square and the associated α and β errors so that they are equal (that is, the ratio is 1). Our model involved 100 DFs and our sample size was N=1000. We defined an unacceptable inappropriate H1 model as one with an RMSEA of at least 0.08. Store the results in a list named cp1.
compromise(effect = .08, effect.measure = 'RMSEA',
N = 1000, df = 100)
The results show that the selection of critical Chi-square =312 is related to the probability of balance error, α= 1.2E-23 and β= 1.2E-23. As required, both of these are equally likely to be errors. If for some reason you want the error probability to be different (for example, because you think that wrongly accepting an incorrect model is 100 times worse than wrongly rejecting a correct model), you can change the ABRATIO parameter. For example, a requirement that the α error be 100 times the β error can be achieved by setting ABRatio = 100.
compromise(effect = .08, effect.measure = 'RMSEA',
abratio = 100, N = 1000, df = 100)
Power Plots
The performance graph shows the relationship between the implicit performance and a number of other variables. You can plot the effectiveness of an effect over a range of sample sizes. Alternatively, you can plot the effectiveness achieved at a given N to detect a range of different effect sizes.
Determine the efficacy of a given effect as a function of N
A graph is created showing the effectiveness achieved in detecting a given effect over a range of sample sizes. However, since it is difficult to specify a diagnostic sample size for a given effect, we need to provide the desired efficacy range. For example, suppose we are interested in how the potency of detecting the effect of RMSEA=.05 changes with N, and we are interested in potency from.05 to.99 (note that the potency cannot be less than α). You can do this by setting the parameters power.min =.05 and power.max =.99. In addition, as with any prior efficacy analysis, you need to define the type and size of the effect, df, and α errors.
powerPlot
This shows that a correlated RMSEA=0.05 model has a very large rejection efficiency at N>250, and a very small rejection efficiency at N<100.
Determine the function relationship between efficiency and effect size at a given N
A graph is created showing the effectiveness achieved over the effect size range for a given sample size. For example, suppose we are interested in how efficacy varies with effect size at N=500, and the corresponding RMSEA range is 0.001 to 0.10. In addition, as with any post-performance analysis, sample size, DF, and α errors need to be defined.
PlotEffect
This shows that at N=500, the potency detected for a related model with RMSEA>0.04 is very large, while the potency detected for RMSEA<0.03 is quite small.
reference
• Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods & Research, 21, 230–258.
• Jöreskog, K. G., & Sörbom, D. (1984). LISREL VI user’s guide (3rd ed.). Mooresville: Scientific Software.
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