logistic_reg_predict
Predict a binomial or multinomial outcome given an estimated model and new values of the independent variables.
Synopsis
#include <imsls.h>
float *imsls_f_logistic_reg_predict (int n_observations, int n_independent, int n_classes, float coefs[], float x[], ..., 0)
The type double function is imsls_d_logistic_reg_predict.
Required Arguments
int n_observations (Input)
The number of observations.
int n_independent (Input)
The number of independent variables.
int n_classes (Input)
The number of discrete outcomes, or classes.
float coefs[] (Input)
Array of length n_coefficients × n_classes containing the coefficient estimates of the logistic regression model. n_coefficients is the number of coefficients in the model.
float x[] (Input)
Array of length n_observations × n_independent containing the values of the independent variables.
Return Value
Pointer to an array containing the predicted responses. The predicted value is the predicted number of outcomes in each class for each new observation provided in x. If frequencies[i] = 1 for all observations, then the return value is equivalent to the predicted probabilities. If the option IMSLS_CONFIDENCE is specified, the length of the return array is (n_observations × n_classes × 3) and the array includes the lower and upper prediction limits. Otherwise, the array is of length (n_observations × n_classes). Note that if the data is column-oriented (see IMSLS_COLUMN_WISE), the return value will also be column‑oriented.
Synopsis with Optional Arguments
#include <imsls.h>
float *imsls_f_logistic_reg_predict (int n_observations, int n_independent, int n_classes, float coefs[], float x[],
IMSLS_Y, floaty[],
IMSLS_GROUP_COUNTS, or
IMSLS_GROUPS,
IMSLS_COLUMN_WISE,
IMSLS_FREQUENCIES, int frequencies[],
IMSLS_REFERENCE_CLASS, int ref_class,
IMSLS_NO_INTERCEPT,
IMSLS_X_INDICES, int n_xin, int xin[],
IMSLS_X_INTERACTIONS, int n_xinteract, int xinteract[],
IMSLS_CONFIDENCE, float confid,
IMSLS_MODEL, Imsls_f_model *model,
IMSLS_AUC, float*auc,
IMSLS_RETURN_USER, floatyhat[],
0)
Optional Arguments
IMSLS_Y, floaty[] (Input)
Array containing the actual responses corresponding to the independent variables. If present, the expected length for y is n_observations × n_classes unless one of IMSLS_GROUPS or IMSLS_GROUP_COUNTS is also present. IMSLS_Y is required when IMSLS_PREDERR is requested.
Default: The function expects that y is not given.
IMSLS_GROUP_COUNTS (Input)
or
IMSLS_GROUPS, (Input)
These optional arguments specify alternative formats of the input array y. If IMSLS_GROUP_COUNTS is present, y is of length n_observations × (n_classes ‑ 1), and contains counts for all but one of the classes for each observation. The missing class is treated as the reference class. If IMSLS_GROUP_COUNTS is present and if any y[i] > 1, IMSLS_FREQUENCIES is required. If IMSLS_GROUPS is present, the input array y is of length n_observations and y[i] contains the group number to which the observation belongs. In this case, frequencies[i] is set to 1 for all observations.
Default: Unless one of the arguments is present, the function expects that y is n_observations × n_classes and contains counts for all the classes.
IMSLS_COLUMN_WISE, (Input)
If present, the input arrays are column‑oriented. That is, contiguous elements in x are values of the same independent variable, or column, except at multiples of n_observations.
Default: Input arrays are row‑oriented.
IMSLS_FREQUENCIES, intfrequencies[] (Input)
Array of length n_observations containing the number of replications or trials for each of the observations. This argument is required if IMSLS_GROUP_COUNTS is present and if any y[i] > 1.
Default: frequencies[i] = 1.
IMSLS_REFERENCE_CLASS, intref_class (Input)
Number specifying which class or outcome category to use as the reference class. The purpose of the reference class is explained in the Description section.
Default: ref_class = n_classes.
IMSLS_NO_INTERCEPT (Input)
If present, the model will not include an intercept term.
Default: The intercept term is included.
IMSLS_X_INDICES, int n_xin, int xin[] (Input)
An array of length n_xin providing the variable indices of x that correspond to the independent variables the user wishes to be included in the logistic regression model.
Default: All n_independent variables are included.
IMSLS_X_INTERACTIONS, int n_xinteract, int xinteract[] (Input)
An array of length n_xinteract × 2 providing pairs of variable indices of x that define the interaction terms in the model. Adjacent indices should be unique.
Default: No interaction terms are included.
IMSLS_CONFIDENCE, floatconfid (Input)
This value provides the confidence level to use in the calculation of the prediction intervals. If this argument is present and valid (0 < confid < 100), confid% prediction intervals are provided for each predicted value.
Default: Prediction intervals are not provided.
IMSLS_MODEL, Imsls_f_model*model (Input)
Pointer to a structure of type Imsls_f_model containing information about the logistic regression fit. See imsls_f_logistic_regression. Required when IMSLS_CONFIDENCE is present.
Default: Not needed if IMSLS_CONFIDENCE is not present.
IMSLS_AUC, float*auc (Output)
The area Under the Curve (AUC) for n_classes = 2 or a multiclass generalization for n_classes > 2 when IMSLS_Y is present.
IMSLS_RETURN_USER, floatyhat[] (Output)
Storage for the return value is provided by the user. See the description of the Return Value above for details.
Description
Function imsls_f_logistic_reg_predict calculates the predicted outcomes for a binomial or multinomial response variable given an estimated logistic regression model and new observations of the independent variables.
For a binary response y, the objective is to estimate the conditional probability of success, 
, where 
is a realization of pindependent variables. In particular, the estimated probability of success
where
and
are the coefficient estimates. Then 
. That is, 
is the expected value of the response under the estimated model given the values of the independent variables.
Similarly, for a multinomial response, with class Kthe reference class,
Then
and 
. If the actual responses are given, the mean squared prediction error is
If requested, 
prediction intervals are provided for the predicted values by first finding the prediction standard errors of the logits, 
, and then evaluating
to obtain the upper and lower limits for 
, where 
is the upper 
quantile of the standard normal distribution. Note that properties of the prediction intervals are only valid when the new observations are inside the range of the original data used to fit the model. Generally, the model should not be used to extrapolate outside the range of the original data. See Hosmer and Lemeshow (2000) for further details.
Examples
Example 1
The model fit to the beetle mortality data of Prentice (1976) is used to predict the expected mortality at three new doses. For the original data, seeExample 1 in imsls_f_logistic_regression.
|
Log Dosage |
Number of Beetles Exposed |
Number of Deaths |
|
1.66 |
16 |
?? |
|
1.87 |
22 |
?? |
|
1.71 |
11 |
?? |
#include <imsls.h>
#include <stdio.h>
int main() {
float y1[8] = { 6, 13, 18, 28, 52, 53, 61, 60 };
float x1[8] = {
1.69, 1.724, 1.755, 1.784, 1.811, 1.836, 1.861, 1.883 };
float x2[3] = { 1.66, 1.87, 1.71 };
float freqs1[8] = { 59, 60, 62, 56, 63, 59, 62, 60 };
float freqs2[3] = { 16, 22, 11 };
float *coefs, *yhat;
int n_classes = 2, n_observations = 8, n_independent = 1,
n_coefs = 2, i, n_new_observations = 3;
coefs = imsls_f_logistic_regression(n_observations, n_independent,
n_classes, x1, y1,
IMSLS_GROUP_COUNTS,
IMSLS_FREQUENCIES, freqs1,
0);
imsls_f_write_matrix("Coefficient Estimates",
n_coefs * (n_classes - 1), 1, coefs, 0);
yhat = imsls_f_logistic_reg_predict(n_new_observations, n_independent,
n_classes, coefs, x2,
IMSLS_FREQUENCIES, freqs2,
0);
printf("\nDose\t N\tExpected Deaths\n");
for(i = 0; i < n_new_observations; i++) {
printf("%5.2f\t%2.1f\t\t%5.2f\n",
x2[i], freqs2[i], yhat[2 * i]);
}
if (coefs)
imsls_free(coefs);
if (yhat)
imsls_free(yhat);
}
Output
Coefficient Estimates
1 -60.76
2 34.30
Dose N Expected Deaths
1.66 16.0 0.34
1.87 22.0 21.28
1.71 11.0 1.19
Example 2
A logistic regression model is fit to artificial (noisy) data with 4 classes and 3 independent variables and used to predict class probabilities at 10 new values of the independent variables. Also shown are the mean squared prediction error and upper and lower limits of the 95% prediction interval for each predicted value.
#include <imsls.h>
#include <stdio.h>
int main() {
float x[50 * 3] = {
3, 2, 2, 1, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 1, 3, 2,
2, 1, 2, 1, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 1, 2, 3, 1, 2,
1, 1, 1, 3, 1, 3, 2, 3, 3, 1,
25.92869, 51.63245, 25.78432, 39.37948, 24.65058, 45.20084,
52.6796, 44.28342, 40.63523, 51.76094, 26.30368, 20.70230,
38.74273, 19.47333, 26.42211, 37.05986, 51.67043, 42.40156,
33.90027, 35.43282, 44.30369, 46.72387, 46.99262, 36.05923,
36.83197, 61.66257, 25.67714, 39.08567, 48.84341, 39.34391,
24.73522, 50.55251, 31.34263, 27.15795, 31.72685, 25.00408,
26.35457, 38.12343, 49.9403, 42.45779, 38.80948, 43.22799,
41.87624, 48.0782, 43.23673, 39.41294, 23.93346,
42.8413, 30.40669, 37.77389,
1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1,
1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2,
1, 1, 2, 1, 1, 2, 1, 1
};
float y[50] = {
1, 2, 3, 4, 3, 3, 4, 4, 4, 4, 2, 1, 4, 1, 1, 1, 4, 4, 3, 1, 2,
3, 3, 4, 2, 3, 4, 1, 2, 4, 3, 4, 4, 1, 3, 4, 4, 2, 3, 4, 2, 2,
4, 3, 1, 4, 3, 4, 2, 3
};
float newx[10 * 3] = {
2, 2, 1, 3, 3, 3, 2, 3, 3, 3,
25.92869, 51.63245, 25.78432, 39.37948, 24.65058, 45.20084,
52.6796, 44.28342, 40.63523, 51.76094,
1, 2, 1, 1, 1, 1, 2, 2, 2, 1
};
float newy[10] = { 3, 2, 1, 1, 4, 3, 2, 2, 1, 2 };
float *coefs, *yhat, auc, model_pval, lrstat;
int i, j, n_classes, n_observations, n_new_obs, n_independent;
int n_coefs, dof;
Imsls_f_model *model_info_ptr = NULL;
n_classes = 4;
n_observations = 50;
n_new_obs = 10;
n_independent = 3;
n_coefs = 4;
coefs = imsls_f_logistic_regression(n_observations, n_independent,
n_classes, x, y,
IMSLS_GROUPS,
IMSLS_COLUMN_WISE,
IMSLS_LRSTAT,&lrstat,
IMSLS_NEXT_RESULTS, &model_info_ptr,
0);
yhat = imsls_f_logistic_reg_predict(n_new_obs, n_independent,
n_classes, coefs, newx,
IMSLS_Y,newy,
IMSLS_GROUPS,
IMSLS_COLUMN_WISE,
IMSLS_CONFIDENCE, 95.0,
IMSLS_MODEL, model_info_ptr,
IMSLS_AUC, &auc,
0);
dof = n_coefs * (n_classes - 1) - (n_classes - 1);
model_pval = 1.0 - imsls_f_chi_squared_cdf(lrstat,dof);
printf("Model Fit Summary:\n");
printf("Log-likelihood: %5.2f \n", model_info_ptr->loglike);
printf("LR test statistic: %5.2f\n", lrstat);
printf("Degrees of freedom: %d\n", dof);
printf("P-value: %5.4f\n", model_pval);
printf("\nPrediction Summary:\n");
printf("Multiclass Area Under the Curve: %4.2f\n", auc);
printf("\nObs Class Estimate Lower Upper\n");
for(j = 0; j < n_new_obs; j++) {
for (i = 0; i < n_classes; i++){
printf(" %d\t%d %4.2f %4.2f %4.2f\n", j+1, i+1,
yhat[i * 3 * n_new_obs + j],
yhat[(i * 3 + 1) * n_new_obs + j],
yhat[(i * 3 + 2) * n_new_obs + j]);
}
}
if (coefs)
imsls_free(coefs);
if (yhat)
imsls_free(yhat);
imsls_f_logistic_regression_free(model_info_ptr);
}
Output
Model Fit Summary:
Log-likelihood: -58.58
LR test statistic: 16.37
Degrees of freedom: 9
P-value: 0.0595
Prediction Summary:
Multiclass Area Under the Curve: 0.56
Obs Class Estimate Lower Upper
1 1 0.26 0.14 0.35
1 2 0.14 0.06 0.20
1 3 0.31 0.18 0.36
1 4 0.29 0.10 0.62
2 1 0.04 0.01 0.14
2 2 0.27 0.11 0.39
2 3 0.12 0.04 0.25
2 4 0.57 0.22 0.85
3 1 0.23 0.07 0.38
3 2 0.13 0.04 0.20
3 3 0.28 0.12 0.34
3 4 0.36 0.08 0.77
4 1 0.06 0.02 0.14
4 2 0.16 0.07 0.24
4 3 0.49 0.28 0.54
4 4 0.29 0.08 0.63
5 1 0.34 0.17 0.41
5 2 0.13 0.06 0.19
5 3 0.30 0.17 0.34
5 4 0.22 0.05 0.60
6 1 0.03 0.00 0.09
6 2 0.16 0.06 0.24
6 3 0.53 0.27 0.60
6 4 0.29 0.07 0.67
7 1 0.04 0.01 0.13
7 2 0.27 0.10 0.40
7 3 0.13 0.04 0.26
7 4 0.57 0.21 0.86
8 1 0.14 0.04 0.26
8 2 0.29 0.12 0.37
8 3 0.12 0.04 0.21
8 4 0.46 0.15 0.80
9 1 0.21 0.08 0.33
9 2 0.27 0.12 0.35
9 3 0.10 0.03 0.19
9 4 0.42 0.14 0.77
10 1 0.01 0.00 0.05
10 2 0.15 0.04 0.24
10 3 0.57 0.23 0.67
10 4 0.28 0.05 0.73
Warning Errors
|
IMSLS_NO_ACTUALS |
The Area Under the Curve (AUC) cannot be calculated because no actual “y” values are given. |
