# response

**Class: **GeneralizedLinearMixedModel

Response vector of generalized linear mixed-effects model

## Description

## 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`

.

## Output Arguments

`y`

— Response values

*n*-by-1 vector

Response values, specified as an *n*-by-1 vector,
where *n* is the number of observations.

For an observation *i* with prior weights *w _{i}^{p}* and
binomial size

*n*(when applicable), the response values

_{i}*y*can have the following values.

_{i}Distribution | Permitted Values | Notes |
---|---|---|

`Binomial` |
$$\left\{0,\frac{1}{{w}_{i}^{p}{n}_{i}},\frac{2}{{w}_{i}^{p}{n}_{i}},\dots ,1\right\}$$ | w and _{i}^{p}n are
integer values > 0_{i} |

`Poisson` |
$$\left\{0,\frac{1}{{w}_{i}^{p}},\frac{2}{{w}_{i}^{p}},\dots \right\}$$ | w is
an integer value > 0_{i}^{p} |

`Gamma` | (0,∞) | w ≥
0_{i}^{p} |

`InverseGaussian` | (0,∞) | w ≥
0_{i}^{p} |

`normal` | (-∞,∞) | w ≥
0_{i}^{p} |

You can access the prior weights property *w _{i}^{p}* using
dot notation. For example, to access the prior weights property for
a model

`glme`

:glme.ObservationInfo.Weights

`binomialsize`

— Binomial size

vector

Binomial size associated with each element of `y`

,
returned as an *n*-by-1 vector, where *n* is
the number of observations. `response`

only returns `binomialsize`

if
the conditional distribution of response given the random effects
is binomial. `binomialsize`

is empty for other
distributions.

## Examples

### Plot Response Versus Fitted Values

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');

Extract the observed response values for the model, then use `fitted`

to generate the fitted conditional mean values.

y = response(glme); % Observed response values yfit = fitted(glme); % Fitted response values

Create a scatterplot of the observed response values versus fitted values. Add a reference line to improve the visualization.

figure scatter(yfit,y) xlim([0,12]) ylim([0,12]) refline(1,0) title('Response versus Fitted Values') xlabel('Fitted Values') ylabel('Response')

The plot shows a positive correlation between the fitted values and the observed response values.

## References

[1] Hox, J. *Multilevel Analysis, Techniques and
Applications*. Lawrence Erlbaum Associates, Inc., 2002.

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