# Log Profile Likelihood of a Parameter

#### Shu Fai Cheung

#### 2023-05-04

Source:`vignettes/loglike.Rmd`

`loglike.Rmd`

## Introduction

This vignette illustrates how to use `loglike_compare()`

in the package `semlbci`

to
compare the log profile likelihood of a parameter in a structural
equation model (SEM) when it is fixed to a range of values to the log
profile likelihood approximated by a quadratic function, using the
method by Pawitan (2013).

## Limitations

The function `loglike_compare()`

and its helper functions
are for creating illustrative examples and learning only, not for
research use. Therefore, they are not as versatile as
`semlbci()`

in the types of models and parameters supported.
They can only be used for free parameters and user-defined parameters
not involved in any constraints. Only a model fitted by maximum
likelihood is supported. Last, they will not check whether the
computation is appropriate for a model. It is the responsibility of the
users to ensure that the computation is appropriate for the model and
parameter.

Despite these limitations, they are still useful for generating graphs for illustration and learning.

## Fitting a Simple Mediation model

Suppose a simple mediation model is fitted to the dataset
`simple_med`

, came with the package `semlbci`

:

```
library(lavaan)
library(semlbci)
dat <- simple_med
mod <-
"
m ~ a * x
y ~ b * m
ab := a * b
"
fit <- lavaan::sem(model = mod,
data = simple_med,
fixed.x = FALSE)
parameterEstimates(fit)
#> lhs op rhs label est se z pvalue ci.lower ci.upper
#> 1 m ~ x a 1.676 0.431 3.891 0.000 0.832 2.520
#> 2 y ~ m b 0.535 0.073 7.300 0.000 0.391 0.679
#> 3 m ~~ m 34.710 3.471 10.000 0.000 27.907 41.513
#> 4 y ~~ y 40.119 4.012 10.000 0.000 32.256 47.982
#> 5 x ~~ x 0.935 0.094 10.000 0.000 0.752 1.118
#> 6 ab := a*b ab 0.897 0.261 3.434 0.001 0.385 1.409
```

## Log Profile likelihood Plots

### The *a*-path

To generate the plot for the *a*-path coefficient, we can
simply call `loglike_compare()`

and setting the parameter to
`"a"`

.

```
ll_a <- loglike_compare(fit,
par_i = "m ~ x")
```

Although teh *a*-path is labelled, `par_i`

requires
the parameter specified in `lavaan::model.syntax`

. Please
refer to the help page of `loglike_compare`

.

By default, 21 values will be used to generate the plot. This can be
controlled by the argument `n_points`

.

Once the run finishes, we can use `plot()`

to plot the
likelihood values:

`plot(ll_a, add_pvalues = TRUE)`

As expected, the log profile likelihood of the *a*-path is
well-approximated by the quadratic approximation.

### The indirect effect

Let us examine the log profile likelihood of the indirect effect. To
specify it, we need to use `ab :=`

:

```
ll_ab <- loglike_compare(fit,
par_i = "ab := ")
```

Note that it usually take longer for parameter which is a nonlinear function of other parameters, such as an indirect effect, which is the product of two other parameters.

This is the plot:

`plot(ll_ab, add_pvalues = TRUE)`

We can see that, as expected, the quadratic approximation is less satisfactory for the indirect effect.

Please refer to the help page of `loglike_compare()`

and
its plot method (`plot.loglike_compare()`

), for other
available options.

## Final Remarks

The other functions, `loglike_range()`

,
`loglike_point()`

, `loglike_quad_range()`

, and
`loglike_quad_point()`

are helper functions used by
`loglike_compare()`

. They are exported such that interested
users can use them to compute the points directly.

More examples can be found in the “examples” folders in the OSF page for this package and Cheung and Pesigan (2023). Some of the figures in the examples were generated by these functions.

## Reference

Cheung, S. F., & Pesigan, I. J. A. (2023). *semlbci*: An R
package for forming likelihood-based confidence intervals for parameter
estimates, correlations, indirect effects, and other derived parameters.
*Structural Equation Modeling: A Multidisciplinary Journal*.
*30*(6), 985–999. https://doi.org/10.1080/10705511.2023.2183860

Pawitan, Y. (2013). *In all likelihood: Statistical modelling and
inference using likelihood*. Oxford University Press.