# fixedEffects

**Class: **GeneralizedLinearMixedModel

Estimates of fixed effects and related statistics

## Syntax

## Description

`[___] = fixedEffects(`

returns
any of the output arguments in previous syntaxes using additional
options specified by one or more `glme`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, you can specify the confidence level, or the
method for computing the approximate degrees of freedom for the *t*-statistic.

## Input Arguments

`glme`

— Generalized linear mixed-effects model

`GeneralizedLinearMixedModel`

object

Generalized linear mixed-effects model, specified as a `GeneralizedLinearMixedModel`

object.
For properties and methods of this object, see `GeneralizedLinearMixedModel`

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`Alpha`

— Significance level

0.05 (default) | scalar value in the range [0,1]

Significance level, specified as the comma-separated pair consisting of
`'Alpha'`

and a scalar value in the range [0,1]. For a value α, the
confidence level is 100 × (1 – α)%.

For example, for 99% confidence intervals, you can specify the confidence level as follows.

**Example: **`'Alpha',0.01`

**Data Types: **`single`

| `double`

`DFMethod`

— Method for computing approximate degrees of freedom

`'residual'`

(default) | `'none'`

Method for computing approximate degrees of freedom, specified
as the comma-separated pair consisting of `'DFMethod'`

and
one of the following.

Value | Description |
---|---|

`'residual'` | The degrees of freedom value is assumed to be constant and equal to n –
p, where n is the number of
observations and p is the number of fixed
effects. |

`'none'` | The degrees of freedom is set to infinity. |

**Example: **`'DFMethod','none'`

## Output Arguments

`beta`

— Estimated fixed-effects coefficients

vector

Estimated fixed-effects coefficients of the fitted generalized
linear mixed-effects model `glme`

, returned as a
vector.

`betanames`

— Names of fixed-effects coefficients

table

Names of fixed-effects coefficients in `beta`

,
returned as a table.

`stats`

— Fixed-effects estimates and related statistics

dataset array

Fixed-effects estimates and related statistics, returned as a dataset array that has one row for each of the fixed effects and one column for each of the following statistics.

Column Name | Description |
---|---|

`Name` | Name of the fixed-effects coefficient |

`Estimate` | Estimated coefficient value |

`SE` | Standard error of the estimate |

`tStat` | t-statistic for a test that the coefficient
is 0 |

`DF` | Estimated degrees of freedom for the t-statistic |

`pValue` | p-value for the t-statistic |

`Lower` | Lower limit of a 95% confidence interval for the fixed-effects coefficient |

`Upper` | Upper limit of a 95% confidence interval for the fixed-effects coefficient |

When fitting a model using `fitglme`

and
one of the maximum likelihood fit methods (`'Laplace'`

or `'ApproximateLaplace'`

),
if you specify the `'CovarianceMethod'`

name-value
pair argument as `'conditional'`

, then `SE`

does
not account for the uncertainty in estimating the covariance parameters.
To account for this uncertainty, specify `'CovarianceMethod'`

as `'JointHessian'`

.

When fitting a GLME model using `fitglme`

and
one of the pseudo likelihood fit methods (`'MPL'`

or `'REMPL'`

), `fixedEffects`

bases
the fixed effects estimates and related statistics on the fitted linear
mixed-effects model from the final pseudo likelihood iteration.

## Examples

### Estimate Fixed-Effects Coefficients

Load the sample data.

`load mfr`

This simulated data is from a manufacturing company that operates 50 factories across the world, with each factory running a batch process to create a finished product. The company wants to decrease the number of defects in each batch, so it developed a new manufacturing process. To test the effectiveness of the new process, the company selected 20 of its factories at random to participate in an experiment: Ten factories implemented the new process, while the other ten continued to run the old process. In each of the 20 factories, the company ran five batches (for a total of 100 batches) and recorded the following data:

Flag to indicate whether the batch used the new process (

`newprocess`

)Processing time for each batch, in hours (

`time`

)Temperature of the batch, in degrees Celsius (

`temp`

)Categorical variable indicating the supplier (

`A`

,`B`

, or`C`

) of the chemical used in the batch (`supplier`

)Number of defects in the batch (

`defects`

)

The data also includes `time_dev`

and `temp_dev`

, which represent the absolute deviation of time and temperature, respectively, from the process standard of 3 hours at 20 degrees Celsius.

Fit a generalized linear mixed-effects model using `newprocess`

, `time_dev`

, `temp_dev`

, and `supplier`

as fixed-effects predictors. Include a random-effects term for intercept grouped by `factory`

, to account for quality differences that might exist due to factory-specific variations. The response variable `defects`

has a Poisson distribution, and the appropriate link function for this model is log. Use the Laplace fit method to estimate the coefficients. Specify the dummy variable encoding as `'effects'`

, so the dummy variable coefficients sum to 0.

The number of defects can be modeled using a Poisson distribution

$${\text{defects}}_{ij}\sim \text{Poisson}({\mu}_{ij})$$

This corresponds to the generalized linear mixed-effects model

$$\mathrm{log}({\mu}_{ij})={\beta}_{0}+{\beta}_{1}{\text{newprocess}}_{ij}+{\beta}_{2}{\text{time}\text{\_}\text{dev}}_{ij}+{\beta}_{3}{\text{temp}\text{\_}\text{dev}}_{ij}+{\beta}_{4}{\text{supplier}\text{\_}\text{C}}_{ij}+{\beta}_{5}{\text{supplier}\text{\_}\text{B}}_{ij}+{b}_{i},$$

where

$${\text{defects}}_{ij}$$ is the number of defects observed in the batch produced by factory $$i$$ during batch $$j$$.

$${\mu}_{ij}$$ is the mean number of defects corresponding to factory $$i$$ (where $$i=1,2,...,20$$) during batch $$j$$ (where $$j=1,2,...,5$$).

$${\text{newprocess}}_{ij}$$, $${\text{time}\text{\_}\text{dev}}_{ij}$$, and $${\text{temp}\text{\_}\text{dev}}_{ij}$$ are the measurements for each variable that correspond to factory $$i$$ during batch $$j$$. For example, $${\text{newprocess}}_{ij}$$ indicates whether the batch produced by factory $$i$$ during batch $$j$$ used the new process.

$${\text{supplier}\text{\_}\text{C}}_{ij}$$ and $${\text{supplier}\text{\_}\text{B}}_{ij}$$ are dummy variables that use effects (sum-to-zero) coding to indicate whether company

`C`

or`B`

, respectively, supplied the process chemicals for the batch produced by factory $$i$$ during batch $$j$$.$${b}_{i}\sim N(0,{\sigma}_{b}^{2})$$ is a random-effects intercept for each factory $$i$$ that accounts for factory-specific variation in quality.

glme = fitglme(mfr,'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)', ... 'Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects');

Compute and display the estimated fixed-effects coefficient values and related statistics.

[beta,betanames,stats] = fixedEffects(glme); stats

stats = Fixed effect coefficients: DFMethod = 'residual', Alpha = 0.05 Name Estimate SE tStat DF pValue {'(Intercept)'} 1.4689 0.15988 9.1875 94 9.8194e-15 {'newprocess' } -0.36766 0.17755 -2.0708 94 0.041122 {'time_dev' } -0.094521 0.82849 -0.11409 94 0.90941 {'temp_dev' } -0.28317 0.9617 -0.29444 94 0.76907 {'supplier_C' } -0.071868 0.078024 -0.9211 94 0.35936 {'supplier_B' } 0.071072 0.07739 0.91836 94 0.36078 Lower Upper 1.1515 1.7864 -0.72019 -0.015134 -1.7395 1.5505 -2.1926 1.6263 -0.22679 0.083051 -0.082588 0.22473

The returned results indicate, for example, that the estimated coefficient for `temp_dev`

is –0.28317. Its large $$p$$-value, 0.76907, indicates that it is not a statistically significant predictor at the 5% significance level. Additionally, the confidence interval boundaries `Lower`

and `Upper`

indicate that the 95% confidence interval for the coefficient for `temp_dev`

is [-2.1926 , 1.6263]. This interval contains 0, which supports the conclusion that `temp_dev`

is not statistically significant at the 5% significance level.

## See Also

`GeneralizedLinearMixedModel`

| `fitglme`

| `coefCI`

| `coefTest`

| `randomEffects`

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