Estimates profile likelihood confidence intervals of predicted group means from a linear or generalized linear model analysis (V.M. Cave).

### Options

`PRINT` = string token |
Controls printed output (`intervals` ); default `inte` |

`BACKTRANSFORM` = string token |
What back-transformation to apply to the intervals on the linear scale (`link` , `none` ); default `link` (i.e. the intervals are presented on the natural scale) |

`PLOT` = string tokens |
What to plot (`profiles` , `intervals` ); default * i.e. no plots |

`DECIMALS` = scalar |
Number of decimal places for printing |

`CIPROBABILITY` = scalar |
Probability for the confidence intervals; default 0.95 |

`RANGE` = scalar, variate or matrix |
Defines the range of values over which to evaluate the profile likelihoods; default 3 |

`NPOINTS` = scalar |
Defines the number of values at which to evaluate the profile likelihoods initially; default 10 |

`NEXTRAPOINTS` = scalar |
Defines the number of extra values to evaluate the profile likelihoods at, within the neighbourhoods of the confidence limits identified during the initial scan as defined by `RANGE` and `NPOINTS` ; default 10 |

`SAVE` = regression save structure |
Regression save structure to provide the information on the regression model; default * uses the most recently fitted regression model |

### Parameter

`CISAVE` = pointer |
Pointer that saves the lower and upper limits of the profile likelihood intervals, along with the estimated group means, labels identifying the groups, and an indicator variable for one-sided intervals |

### Description

`RMPLCONFIDENCE`

estimates profile likelihood confidence intervals of group means from a linear or generalized linear model analysis. The model must have been fitted already using the regression commands `MODEL`

and `FIT`

in the usual way.

The groups are defined by the unique combinations of the factors in the regression model. Therefore, the model formula (defined by the `FIT`

directive) must contain at least one factor, and if more than one factor is included, the model formula must represent the full factorial of these factors. Furthermore, the model formula must not contain covariates.

By default, the confidence intervals are produced for the most recent regression analysis. However, you can use the `SAVE`

option to specify the save structure from another regression analysis (see the `SAVE`

option of `MODEL`

).

The probability for the confidence intervals is specified by the `CIPROBABILITY`

option; the default 0.95 gives 95% confidence intervals.

To estimate the profile likelihood confidence interval of a group’s mean, `RMPLCONFIDENCE`

refits the model with the group’s mean fixed at a given value, and then computes the change in deviance between this model and the original model, in which the group’s mean has been estimated as per normal. This process is repeated across a range of values for the group mean, aimed to be broad enough to include its lower and upper confidence limits. These changes in deviance are then compared to the critical value for the `CIPROBABILITY`

× 100th percentile of the chi-square distribution on 1 degree of freedom, and the confidence limits are estimated by interpolating between the fixed values for the group mean that resulted in deviance changes closest to this critical value. The procedure uses the `INTERPOLATE`

directive to perform cubic interpolation, or linear interpolation if cubic interpolation is not possible.

The range of values over which `RMPLCONFIDENCE`

evaluates the profile likelihood of a group’s mean is defined by the option `RANGE`

, in either a scalar, variate or matrix. If a scalar is supplied, this specifies a multiplier of the standard error of the mean, and points are evaluated at this multiple of the standard error, in both the positive and negative directions from the estimated group mean. Alternatively, if a variate is supplied, it must contain two values, specifying the deviations in the negative and positive directions from the estimated mean, respectively. Finally, if a matrix is supplied, it must have two columns and the same number of rows as group means. The value in the first column of the *i*th row specifies the deviation in the negative direction from the estimated group mean *i*, and the value in the second column of the *i*th row specifies the deviation in the positive direction from the estimated group mean *i*. The default of three evaluates the profile likelihood at points ranging up to three times the standard error of the mean in both the negative and positive directions. In general, this default works well. However, for very skewed or flat profile likelihoods, a more appropriate `RANGE`

will need to be specified.The `NPOINTS`

option specifies the number of points where the profile likelihood is evaluated initially, in both the negative and positive directions from the estimated mean. The default is 10. That is, 10 evenly spaced values between the lower bound defined by `RANGE`

, up to (but not including) the estimated mean, are evaluated in order to estimate the lower limit of the confidence interval. Similarly, 10 evenly spaced values between the upper bound defined by `RANGE`

, down to (but not including) the estimated mean, are evaluated in order to estimate the upper limit of the confidence interval.

To get more accurate estimates of the confidence limits, a finer search of the profile likelihood can be performed by setting the option `NEXTRAPOINTS`

to an integer greater than zero. The profile likelihood at `NEXTRAPOINTS`

is then evaluated around the neighbourhoods of both the lower and upper confidence limits identified during the initial search (as defined by `RANGE`

and `NPOINTS`

). For example, if the initial search identified that the lower limit was located between the values of 1 and 3, `NEXTRAPOINTS=4`

would cause the profile likelihood to be further evaluated at 1.4, 1.8, 2.2, 2.6. Setting `NEXTRAPOINTS=0`

means a finer scan is not performed; default 10.

Printed output is controlled by the `PRINT`

option. By default, `RMPLCONFIDENCE`

prints the profile likelihood confidence intervals, but you can suppress this by setting option `PRINT=*`

. The number of decimals places to appear in the printed output can be set using the `DECIMALS`

option.

The `PLOT`

option allows you to request plots of the results, using the following settings.

`profiles`

for plots of the profile likelihoods for each group mean. Superimposed onto these plots are a horizontal reference line at the critical value (in red), vertical projections from the critical value to the estimated confidence limits (also in red) and a vertical reference line at the estimated mean (green dotted). Blue crosses on the profile plot indicate fixed values for the group mean at which the regression model failed to successfully fit.

`intervals`

for a plot of the estimated profile likelihood confidence intervals of the group means. By default, no plots are produced.

The `CISAVE`

parameter can supply a pointer to store the results. The elements of the pointer are labelled as follows:

`'label'`

text structure storing labels to identify the group means,

`'estimate'`

variate storing the estimated group means,

`'lower'`

variate storing the lower limits of the profile likelihood intervals,

`'upper'`

variate storing the upper limits of the profile likelihood intervals,

`'one-sided'`

indicator variable identifying one-sided intervals.

A one-sided interval is produced when the estimated mean lies at the bound of the assumed distribution. Thus, for regression models with a Poisson, geometric or negative binomial distribution, right-sided intervals are produced for groups whose estimated mean is 0 (i.e. at the lower bound of the distribution). For models with binomial or Bernoulli distributions, left- and right-sided intervals are produced when the number of failures, or the number of successes, is the same as the number of trials, respectively (i.e. when the estimated proportion for a group is 0 or 1).

`RMPLCONFIDENCE`

prints, plots and saves the profile confidence intervals on the back-transformed scale (i.e. on the natural scale). However, you can obtain the intervals on the scale of the linear predictor by setting the option `BACKTRANSFORM=none`

.

The procedure accommodates most linear and generalized linear models defined by the `MODEL`

directive, except for models with a user defined (i.e. `calculated`

) `DISTRIBUTION`

or `LINK`

function, a multinomial distribution or an absorbing factor (i.e. when the `GROUPS`

option is used). In addition, although the model formula defined by the `FIT`

directive may include `POL`

, REG and `COMPARISON`

functions, these are ignored by the procedure.

Options: `PRINT`

, `BACKTRANSFORM`

, `PLOT`

, `DECIMALS`

, `CIPROBABILITY`

, `RANGE`

, `NPOINTS`

, `NEXTRAPOINTS`

, `SAVE`

Parameter: `CISAVE`

.

### See also

Directives: `MODEL`

, `FIT`

.

Procedure: `BACKTRANSFORM`

, `RPLCONFIDENCE`

.

Commands for: Regression analysis.

### Example

CAPTION 'RMPLCONFIDENCE examples'; STYLE=meta " Linear regression model " SPLOAD '%Data%/Calcium.gsh' MODEL calcium FIT region RMPLCONFIDENCE [PLOT=profiles,intervals] " GLM: log-linear model " SPLOAD '%Data%/Pups.gsh' MODEL [DISTRIBUTION=Poisson; LINK=log] pups FIT diet*time RMPLCONFIDENCE [PLOT=profiles,intervals]