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| --- | ||
| title: 'Anova' | ||
| output: | ||
| bookdown::tufte_html2 | ||
| --- | ||
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| # Anova | ||
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| ## Anova for between subjects designs | ||
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| TODO | ||
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| ## Repeated measures | ||
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| TODO | ||
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| Original file line number | Diff line number | Diff line change |
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| --- | ||
| title: 'Confirmatory factor analysis' | ||
| output: | ||
| bookdown::tufte_html2 | ||
| --- | ||
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| # Confirmatory factor analysis | ||
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| *This section adapted from a guide originally produced by Jon May* | ||
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| First make sure the `lavaan` package is installed^[see: [installing packages](packages.html)]; it stands for Latent Variable Analysis and is the most popular package for CFA in R. | ||
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| ```r | ||
| install.packages(lavaan) | ||
| ``` | ||
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| ```r | ||
| library(lavaan) | ||
| ``` | ||
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| Open your data and check that all looks well: | ||
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| ```r | ||
| hz <- lavaan::HolzingerSwineford1939 | ||
| hz %>% glimpse() | ||
| ## Observations: 301 | ||
| ## Variables: 15 | ||
| ## $ id <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, ... | ||
| ## $ sex <int> 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2... | ||
| ## $ ageyr <int> 13, 13, 13, 13, 12, 14, 12, 12, 13, 12, 12, 12, 12, 12,... | ||
| ## $ agemo <int> 1, 7, 1, 2, 2, 1, 1, 2, 0, 5, 2, 11, 7, 8, 6, 1, 11, 5,... | ||
| ## $ school <fctr> Pasteur, Pasteur, Pasteur, Pasteur, Pasteur, Pasteur, ... | ||
| ## $ grade <int> 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7... | ||
| ## $ x1 <dbl> 3.333333, 5.333333, 4.500000, 5.333333, 4.833333, 5.333... | ||
| ## $ x2 <dbl> 7.75, 5.25, 5.25, 7.75, 4.75, 5.00, 6.00, 6.25, 5.75, 5... | ||
| ## $ x3 <dbl> 0.375, 2.125, 1.875, 3.000, 0.875, 2.250, 1.000, 1.875,... | ||
| ## $ x4 <dbl> 2.333333, 1.666667, 1.000000, 2.666667, 2.666667, 1.000... | ||
| ## $ x5 <dbl> 5.75, 3.00, 1.75, 4.50, 4.00, 3.00, 6.00, 4.25, 5.75, 5... | ||
| ## $ x6 <dbl> 1.2857143, 1.2857143, 0.4285714, 2.4285714, 2.5714286, ... | ||
| ## $ x7 <dbl> 3.391304, 3.782609, 3.260870, 3.000000, 3.695652, 4.347... | ||
| ## $ x8 <dbl> 5.75, 6.25, 3.90, 5.30, 6.30, 6.65, 6.20, 5.15, 4.65, 4... | ||
| ## $ x9 <dbl> 6.361111, 7.916667, 4.416667, 4.861111, 5.916667, 7.500... | ||
| ``` | ||
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| ## Defining the model | ||
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| Define the CFA model you want to run using 'lavaan model syntax'^[A full guide is here: <http://lavaan.ugent.be/tutorial/syntax1.html>]. For example: | ||
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| ```r | ||
| hz.model <- ' | ||
| visual =~ x1 + x2 + x3 | ||
| writing =~ x4 + x5 + x6 | ||
| maths =~ x7 + x8 + x9 | ||
| ' | ||
| ``` | ||
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| The model is defined in text, and can be broken over lines for clarity. Here was save it in a variable named `hz.model`. | ||
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| The model is defined by naming latent variables, and then specifying which observed variables measure them. The latent variable names are followed by =~ which means 'is manifested by'. | ||
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| To run the analysis: | ||
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| ```r | ||
| hz.fit <- cfa(hz.model, data=hz) | ||
| ``` | ||
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| To display the results: | ||
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| ```r | ||
| hz.fit.summary <- summary(hz.fit, standardized=TRUE) | ||
| ## lavaan (0.5-23.1097) converged normally after 35 iterations | ||
| ## | ||
| ## Number of observations 301 | ||
| ## | ||
| ## Estimator ML | ||
| ## Minimum Function Test Statistic 85.306 | ||
| ## Degrees of freedom 24 | ||
| ## P-value (Chi-square) 0.000 | ||
| ## | ||
| ## Parameter Estimates: | ||
| ## | ||
| ## Information Expected | ||
| ## Standard Errors Standard | ||
| ## | ||
| ## Latent Variables: | ||
| ## Estimate Std.Err z-value P(>|z|) Std.lv Std.all | ||
| ## visual =~ | ||
| ## x1 1.000 0.900 0.772 | ||
| ## x2 0.554 0.100 5.554 0.000 0.498 0.424 | ||
| ## x3 0.729 0.109 6.685 0.000 0.656 0.581 | ||
| ## writing =~ | ||
| ## x4 1.000 0.990 0.852 | ||
| ## x5 1.113 0.065 17.014 0.000 1.102 0.855 | ||
| ## x6 0.926 0.055 16.703 0.000 0.917 0.838 | ||
| ## maths =~ | ||
| ## x7 1.000 0.619 0.570 | ||
| ## x8 1.180 0.165 7.152 0.000 0.731 0.723 | ||
| ## x9 1.082 0.151 7.155 0.000 0.670 0.665 | ||
| ## | ||
| ## Covariances: | ||
| ## Estimate Std.Err z-value P(>|z|) Std.lv Std.all | ||
| ## visual ~~ | ||
| ## writing 0.408 0.074 5.552 0.000 0.459 0.459 | ||
| ## maths 0.262 0.056 4.660 0.000 0.471 0.471 | ||
| ## writing ~~ | ||
| ## maths 0.173 0.049 3.518 0.000 0.283 0.283 | ||
| ## | ||
| ## Variances: | ||
| ## Estimate Std.Err z-value P(>|z|) Std.lv Std.all | ||
| ## .x1 0.549 0.114 4.833 0.000 0.549 0.404 | ||
| ## .x2 1.134 0.102 11.146 0.000 1.134 0.821 | ||
| ## .x3 0.844 0.091 9.317 0.000 0.844 0.662 | ||
| ## .x4 0.371 0.048 7.779 0.000 0.371 0.275 | ||
| ## .x5 0.446 0.058 7.642 0.000 0.446 0.269 | ||
| ## .x6 0.356 0.043 8.277 0.000 0.356 0.298 | ||
| ## .x7 0.799 0.081 9.823 0.000 0.799 0.676 | ||
| ## .x8 0.488 0.074 6.573 0.000 0.488 0.477 | ||
| ## .x9 0.566 0.071 8.003 0.000 0.566 0.558 | ||
| ## visual 0.809 0.145 5.564 0.000 1.000 1.000 | ||
| ## writing 0.979 0.112 8.737 0.000 1.000 1.000 | ||
| ## maths 0.384 0.086 4.451 0.000 1.000 1.000 | ||
| hz.fit.summary | ||
| ## NULL | ||
| ``` | ||
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| The output has three parts: | ||
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| 1. Parameter estimates. The values in the first column are the standardised weights from the observed variables to the latent factors. | ||
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| 2. Factor covariances. The values in the first column are the covariances between the latent factors. | ||
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| 3. Error variances. The values in the first column are the estimates of each observed variable’s error variance. | ||
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| ## Model fit | ||
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| To examine the model fit: | ||
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| ```r | ||
| fitmeasures(hz.fit, c('cfi', 'rmsea', 'rmsea.ci.upper', 'bic')) | ||
| ## cfi rmsea rmsea.ci.upper bic | ||
| ## 0.931 0.092 0.114 7595.339 | ||
| ``` | ||
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| This looks OK, but could be improved. | ||
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| ## Modification indices | ||
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| To examine the modification indices (here sorted to see the largest first) we type: | ||
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| ```r | ||
| modificationindices(hz.fit) %>% | ||
| as.data.frame() %>% | ||
| arrange(-mi) %>% | ||
| filter(mi > 10) %>% | ||
| select(lhs, op, rhs, mi, epc) %>% | ||
| pander::pander() | ||
| ``` | ||
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| ------------------------------- | ||
| lhs op rhs mi epc | ||
| ------ ---- ----- ----- ------- | ||
| visual =~ x9 36.41 0.577 | ||
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| x7 ~~ x8 34.15 0.5364 | ||
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| visual =~ x7 18.63 -0.4219 | ||
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| x8 ~~ x9 14.95 -0.4231 | ||
| ------------------------------- | ||
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| Latent factor to variable links have `=~` in the 'op' column. Error covariances for observed variables have `~~` as the op. These symbols match the symbols used to describe a path in the lavaan model syntax. | ||
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| If we add the largest MI path to our model it will look like this: | ||
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| ```r | ||
| hz.model.2 <- " | ||
| visual =~ x1 + x2 + x3 | ||
| writing =~ x4 + x5 + x6 + x9 | ||
| maths =~ x7 + x8 + x9 | ||
| " | ||
| hz.fit.2 <- cfa(hz.model.2, data=hz) | ||
| fitmeasures(hz.fit.2, c('cfi', 'rmsea', 'rmsea.ci.upper', 'bic')) | ||
| ## cfi rmsea rmsea.ci.upper bic | ||
| ## 0.936 0.091 0.113 7595.660 | ||
| ``` | ||
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| RMSEA has improved marginally, but we'd probably want to investigate this model further, and make additional improvements to it. | ||
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| ## Missing data | ||
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| If you have missing data, add the argument `estimator='MLM'` to the `cfa()` function to use a robust method. There are no missing data in this dataset, but it would look like this: | ||
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| ```r | ||
| hz.fit.2.mlm <- cfa(hz.model.2, data=hz, estimator="MLM") | ||
| hz.fit.2.mlm | ||
| ## lavaan (0.5-23.1097) converged normally after 35 iterations | ||
| ## | ||
| ## Number of observations 301 | ||
| ## | ||
| ## Estimator ML Robust | ||
| ## Minimum Function Test Statistic 79.919 75.703 | ||
| ## Degrees of freedom 23 23 | ||
| ## P-value (Chi-square) 0.000 0.000 | ||
| ## Scaling correction factor 1.056 | ||
| ## for the Satorra-Bentler correction | ||
| ``` | ||
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