summary.laplace.ppc {LaplacesDemon}

Posterior Predictive Check Summary

Description

This may be used to summarize either new, unobserved instances of \textbf{y} (called \textbf{y}^{new}) or replicates of \textbf{y} (called \textbf{y}^{rep}). Either \textbf{y}^{new} or \textbf{y}^{rep} is summarized, depending on predict.laplace.

Usage

## S3 method for class 'laplace.ppc'
summary(object, Categorical, Rows, Discrep,
     d, Quiet, ...)

Arguments

object

An object of class laplace.ppc is required.

Categorical

Logical. If TRUE, then y and yhat are considered to be categorical (such as y=0 or y=1), rather than continuous.

Rows

An optional vector of row numbers, for example c(1:10). All rows will be estimated, but only these rows will appear in the summary.

Discrep

A character string indicating a discrepancy test. Discrep defaults to NULL. Valid character strings when y is continuous are: "Chi-Square", "Chi-Square2", "Kurtosis", "L-criterion", "MASE", "MSE", "PPL", "Quadratic Loss", "Quadratic Utility", "RMSE", "Skewness", "max(yhat[i,]) > max(y)", "mean(yhat[i,]) > mean(y)", "mean(yhat[i,] > d)", "mean(yhat[i,] > mean(y))", "min(yhat[i,]) < min(y)", "round(yhat[i,]) = d", and "sd(yhat[i,]) > sd(y)". Valid character strings when y is categorical are: "p(yhat[i,] != y[i])". Kurtosis and skewness are not discrepancies, but are included here for convenience.

d

This is an optional integer to be used with the Discrep argument above, and it defaults to d=0.

Quiet

This logical argument defaults to FALSE and will print results to the console. When TRUE, results are not printed.

...

Additional arguments are unused.

Details

This function summarizes an object of class laplace.ppc, which consists of posterior predictive checks on either \textbf{y}^{new} or \textbf{y}^{rep}, depending respectively on whether unobserved instances of \textbf{y} or the model sample of \textbf{y} was used in the predict.laplace function. The deviance and monitored variables are also summarized.

The purpose of a posterior predictive check is to assess how well (or poorly) the model fits the data, or to assess discrepancies between the model and the data. For more information on posterior predictive checks, see https://web.archive.org/web/20150215050702/http://www.bayesian-inference.com/posteriorpredictivechecks.

When \textbf{y} is continuous and known, this function estimates the predictive concordance between \textbf{y} and \textbf{y}^{rep} as per Gelfand (1996), and the predictive quantile (PQ), which is for record-level outlier detection used to calculate Gelfand's predictive concordance.

When \textbf{y} is categorical and known, this function estimates the record-level lift, which is p(yhat[i,] = y[i]) / [p(y = j) / n], or the number of correctly predicted samples over the rate of that category of \textbf{y} in vector \textbf{y}.

A discrepancy measure is an approach to studying discrepancies between the model and data (Gelman et al., 1996). Below is a list of discrepancy measures, followed by a brief introduction to discrepancy analysis:

• The "Chi-Square" discrepancy measure is the chi-square goodness-of-fit test that is recommended by Gelman. For each record i=1:N, this returns (y[i] - E(y[i]))^2 / var(yhat[i,]).

• The "Chi-Square2" discrepancy measure returns the following for each record: Pr(chisq.rep[i,] > chisq.obs[i,]), where chisq.obs[i,] <- (y[i] - E(y[i]))^2 / E(y[i]), and chisq.rep[i,] <- (yhat[i,] - E(yhat[i,]))^2 / E(yhat[i,]), and the overall discrepancy is the percent of records that were outside of the 95% quantile-based probability interval (see p.interval).

• The "Kurtosis" discrepancy measure returns the kurtosis of \textbf{y}^{rep} for each record, and the discrepancy statistic is the mean for all records. This does not measure discrepancies between the model and data, and is useful for finding kurtotic replicate distributions.

• The "L-criterion" discrepancy measure of Laud and Ibrahim (1995) provides the record-level combination of two components (see below), and the discrepancy statistic is the sum, L, as well as a calibration number, S.L. For more information on the L-criterion, see the accompanying vignette entitled "Bayesian Inference".

• The "MASE" (Mean Absolute Scaled Error) is a discrepancy measure for the accuracy of time-series forecasts, estimated as (|y - yhat|) / mean(abs(diff(y))). The discrepancy statistic is the mean of the record-level values.

• The "MSE" (Mean Squared Error) discrepancy measure provides the MSE for each record across all replicates, and the discrepancy statistic is the mean of the record-level MSEs. MSE and quadratic loss are identical.

• The "PPL" (Posterior Predictive Loss) discrepancy measure of Gelfand and Ghosh (1998) provides the record-level combination of two components: one involves the predictive variance and the other includes the accuracy of the means of the predictive distribution. The d=0 argument applies the following weight to the accuracy component, which is then added to the variance component: d/(d+1). For \textbf{y}^{new}, use d=0. For \textbf{y}^{rep} and model comparison, d is commonly set to 1, 10, or 100000. Larger values of d put more stress on fit and downgrade the precision of the estimates.

• The "Quadratic Loss" discrepancy measure provides the mean quadratic loss for each record across all replicates, and the discrepancy statistic is the mean of the record-level mean quadratic losses. Quadratic loss and MSE are identical, and quadratic loss is the negative of quadratic utility.

• The "Quadratic Utility" discrepancy measure provides the mean quadratic utility for each record across all replicates, and the discrepancy statistic is the mean of the record-level mean quadratic utilities. Quadratic utility is the negative of quadratic loss.

• The "RMSE" (Root Mean Squared Error) discrepancy measure provides the RMSE for each record across all replicates, and the discrepancy statistic is the mean of the record-level RMSEs.

• The "Skewness" discrepancy measure returns the skewness of \textbf{y}^{rep} for each record, and the discrepancy statistic is the mean for all records. This does not measure discrepancies between the model and data, and is useful for finding skewed replicate distributions.

• The "max(yhat[i,]) > max(y)" discrepancy measure returns a record-level indicator when a record's maximum \textbf{y}^{rep}_i exceeds the maximum of \textbf{y}. The discrepancy statistic is the mean of the record-level indicators, reporting the proportion of records with replications that exceed the maximum of \textbf{y}.

• The "mean(yhat[i,]) > mean(y)" discrepancy measure returns a record-level indicator when the mean of a record's \textbf{y}^{rep}_i is greater than the mean of \textbf{y}. The discrepancy statistic is the mean of the record-level indicators, reporting the proportion of records with mean replications that exceed the mean of \textbf{y}.

• The "mean(yhat[i,] > d)" discrepancy measure returns a record-level proportion of \textbf{y}^{rep}_i that exceeds a specified value, d. The discrepancy statistic is the mean of the record-level proportions.

• The "mean(yhat[i,] > mean(y))" discrepancy measure returns a record-level proportion of \textbf{y}^{rep}_i that exceeds the mean of \textbf{y}. The discrepancy statistic is the mean of the record-level proportions.

• The "min(yhat[i,]) < min(y)" discrepancy measure returns a record-level indicator when a record's minimum \textbf{y}^{rep}_i is less than the minimum of \textbf{y}. The discrepancy statistic is the mean of the record-level indicators, reporting the proportion of records with replications less than the minimum of \textbf{y}.

• The "round(yhat[i,]) = d" discrepancy measure returns a record-level proportion of \textbf{y}^{rep}_i that, when rounded, is equal to a specified discrete value, d. The discrepancy statistic is the mean of the record-level proportions.

• The "sd(yhat[i,]) > sd(y)" discrepancy measure returns a record-level indicator when the standard deviation of replicates is larger than the standard deviation of all of \textbf{y}. The discrepancy statistic is the mean of the record-level indicators, reporting the proportion of records with larger standard deviations than \textbf{y}.

• The "p(yhat[i,] != y[i])" discrepancy measure returns the record-level probability that \textbf{y}^{rep}_i is not equal to \textbf{y}. This is valid when \textbf{y} is categorical and yhat is the predicted category. The probability is the proportion of replicates.

After observing a discrepancy statistic, the user attempts to improve the model by revising the model to account for discrepancies between data and the current model. This approach to model revision relies on an analysis of the discrepancy statistic. Given a discrepancy measure that is based on model fit, such as the L-criterion, the user may correlate the record-level discrepancy statistics with the dependent variable, independent variables, and interactions of independent variables. The discrepancy statistic should not correlate with the dependent and independent variables. Interaction variables may be useful for exploring new relationships that are not in the current model. Alternatively, a decision tree may be applied to the record-level discrepancy statistics, given the independent variables, in an effort to find relationships in the data that may be helpful in the model. Model revision may involve the addition of a finite mixture component to account for outliers in discrepancy, or specifying the model with a distribution that is more robust to outliers. There are too many suggestions to include here, and discrepancy analysis varies by model.

Value

This function returns a list with the following components:

BPIC

The Bayesian Predictive Information Criterion (BPIC) was introduced by Ando (2007). BPIC is a variation of the Deviance Information Criterion (DIC) that has been modified for predictive distributions. For more information on DIC (Spiegelhalter et al., 2002), see the accompanying vignette entitled "Bayesian Inference". BPIC = Dbar + 2pD. The goal is to minimize BPIC.

Concordance

This is the percentage of the records of y that are within the 95% quantile-based probability interval (see p.interval) of \textbf{y}^{rep}. Gelfand's suggested goal is to achieve 95% predictive concordance. Lower percentages indicate too many outliers and a poor fit of the model to the data, and higher percentages may suggest overfitting. Concordance occurs only when \textbf{y} is continuous.

Mean Lift

This is the mean of the record-level lifts, and occurs only when \textbf{y} is specified as categorical with Categorical=TRUE.

Discrepancy.Statistic

This is only reported if the Discrep argument receives a valid discrepancy measure as listed above. The Discrep applies to each record of \textbf{y}, and the Discrepancy.Statistic reports the results of the discrepancy measure on the entire data set. For example, if Discrep="min(yhat[i,]) < min(y)", then the overall result is the proportion of records in which the minimum sample of yhat was less than the overall minimum \textbf{y}. This is Pr(min(yhat[i,]) < min(y) | y, Theta), where Theta is the parameter set.

L-criterion

The L-criterion (Laud and Ibrahim, 1995) was developed for model and variable selection. It is a sum of two components: one involves the predictive variance and the other includes the accuracy of the means of the predictive distribution. The L-criterion measures model performance with a combination of how close its predictions are to the observed data and variability of the predictions. Better models have smaller values of L. L is measured in the same units as the response variable, and measures how close the data vector \textbf{y} is to the predictive distribution. In addition to the value of L, there is a value for S.L, which is the calibration number of L, and is useful in determining how much of a decrease is necessary between models to be noteworthy.

Monitor

This is a N \times 5 matrix, where N is the number of monitored variables and there are 5 columns, as follows: Mean, SD, LB (the 2.5% quantile), Median, and UB (the 97.5% quantile).

Summary

When \textbf{y} is continuous, this is a N \times 8 matrix, where N is the number of records of \textbf{y} and there are 8 columns, as follows: y, Mean, SD, LB (the 2.5% quantile), Median, UB (the 97.5% quantile), PQ (the predictive quantile, which is Pr(\textbf{y}^{rep} \ge \textbf{y})), and Test, which shows the record-level result of a test, if specified. When \textbf{y} is categorical, this matrix has a number of columns equal to the number of categories of \textbf{y} plus 3, also including y, Lift, and Discrep.

Author(s)

Statisticat, LLC.

References

Ando, T. (2007). "Bayesian Predictive Information Criterion for the Evaluation of Hierarchical Bayesian and Empirical Bayes Models". Biometrika, 94(2), p. 443–458.

Gelfand, A. (1996). "Model Determination Using Sampling Based Methods". In Gilks, W., Richardson, S., Spiegehalter, D., Chapter 9 in Markov Chain Monte Carlo in Practice. Chapman and Hall: Boca Raton, FL.

Gelfand, A. and Ghosh, S. (1998). "Model Choice: A Minimum Posterior Predictive Loss Approach". Biometrika, 85, p. 1–11.

Gelman, A., Meng, X.L., and Stern H. (1996). "Posterior Predictive Assessment of Model Fitness via Realized Discrepancies". Statistica Sinica, 6, p. 733–807.

Laud, P.W. and Ibrahim, J.G. (1995). "Predictive Model Selection". Journal of the Royal Statistical Society, B 57, p. 247–262.

Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A. (2002). "Bayesian Measures of Model Complexity and Fit (with Discussion)". Journal of the Royal Statistical Society, B 64, p. 583–639.

See Also

LaplaceApproximation, predict.laplace, and p.interval.

Examples

### See the LaplaceApproximation function for an example.