The GAM Procedure

Example 41.2 Poisson Regression Analysis of Component Reliability

In this example, the number of maintenance repairs on a complex system are modeled as realizations of Poisson random variables. The system under investigation has a large number of components, which occasionally break down and are replaced or repaired. During a four-year period, the system was observed to be in a state of steady operation, meaning that the rate of operation remained approximately constant. A monthly maintenance record is available for that period, which tracks the number of components removed for maintenance each month. The data are listed in the following statements, which create a SAS data set:

title 'Analysis of Component Reliability';
data equip;
   input year month removals @@;
1987   1  2 1987   2  4 1987   3  3
1987   4  3 1987   5  3 1987   6  8
1987   7  2 1987   8  6 1987   9  3
1987  10  9 1987  11  4 1987  12 10
1988   1  4 1988   2  6 1988   3  4
1988   4  4 1988   5  3 1988   6  5
1988   7  3 1988   8  4 1988   9  5
1988  10  3 1988  11  6 1988  12  3
1989   1  2 1989   2  6 1989   3  1
1989   4  5 1989   5  5 1989   6  4
1989   7  2 1989   8  2 1989   9  2
1989  10  5 1989  11  1 1989  12 10
1990   1  3 1990   2  8 1990   3 12
1990   4  7 1990   5  3 1990   6  2
1990   7  4 1990   8  3 1990   9  0
1990  10  6 1990  11  6 1990  12  6

For planning purposes, it is of interest to understand the long- and short-term trends in the maintenance needs of the system. Over the long term, it is suspected that the quality of new components and repair work improves over time, so the number of component removals would tend to decrease from year to year. It is not known whether the robustness of the system is affected by seasonal variations in the operating environment, but this possibility is also of interest.

Because the maintenance record is in the form of counts, the number of removals are modeled as realizations of Poisson random variables. Denote by $\lambda _{ij}$ the unobserved component removal rate for year i and month j. Since the data were recorded at regular intervals (from a system operating at a constant rate), each $\lambda _{ij}$ is assumed to be a function of year and month only.

A preliminary two-way analysis is performed by using PROC GENMOD to make broad inferences on repair trends. A log-link is specified for the model

\begin{eqnarray*}  \log \lambda _{ij} &  = &  \mu + \alpha ^ Y_ i + \alpha ^ M_ j \end{eqnarray*}

where $\mu $ is a grand mean, $\alpha ^ Y_ i$ is the effect of the ith year, and $\alpha ^ M_ j$ is the effect of the jth month.

In the following statements, the CLASS statement declares the variables year and month as categorical. Type III sum of squares are requested to test whether there is an overall effect of year and/or month.

title2 'Two-way model';
proc genmod data=equip;
   class year month;
   model removals=year month / dist=Poisson link=log type3;

Output 41.2.1 displays the listed Type III statistics for the fitted model. With the test for year effects yielding a p-value of 0.4527, there is no evidence of a long-term trend in maintenance rates. Apparently, the quality of new or repaired components did not change between 1987 and 1990. However, the test for monthly trends does yield a small p-value of 0.0321, indicating that seasonal trends are significant at the $\alpha =0.05$ level.

Output 41.2.1: PROC GENMOD Listing for Type III Analysis

Analysis of Component Reliability
Two-way model

The GENMOD Procedure

LR Statistics For Type 3 Analysis
Source DF Chi-Square Pr > ChiSq
year 3 2.63 0.4527
month 11 21.12 0.0321

If year is dropped from the model, the focus of the analysis is now on identifying the form of the underlying seasonal trend, which is a task that PROC GAM is especially suited for. PROC GAM will be used to fit both a reduced categorical model, with year eliminated, and a nonparametric spline model. Although PROC GENMOD also has the capability to fit categorical models, as demonstrated earlier, PROC GAM will be used here to fit both models for a better comparison.

The following PROC GAM statements specify the reduced categorical model and write predicted values to a data set. For this part of the analysis, a CLASS statement is again used to specify that month is a categorical variable. In the follow-up, the seasonal effect will be treated as a nonparametric function of month.

title2 'One-way model';
proc gam data=equip;
   class month;
   model removals=param(month) / dist=Poisson;
   output out=est p;

The following statements use the SGPLOT procedure to generate a plot of the estimated seasonal trend. The plot is displayed in Output 41.2.2.

proc sort data=est;by month;run;
proc sgplot data=est;
   title "Predicted Seasonal Trend";
   yaxis label="Number of Removals";
   xaxis integer values=(1 to 12);
   scatter x=Month y=Removals / name="points"
   series  x=Month y=p_Removals / name="line"
                                  legendLabel="Predicted Removals"
                                  lineattrs = GRAPHFIT;
   discretelegend "points" "line";

Output 41.2.2: Predicted Seasonal Trend from a Parametric Model Fit Using a CLASS Statement

Predicted Seasonal Trend from a Parametric Model Fit Using a CLASS Statement

The predicted repair rates shown in Output 41.2.2 form a jagged seasonal pattern. Ignoring the month-to-month fluctuations, which are difficult to explain and can be artifacts of random noise, the general removal rate trend is high in winter and low in summer.

One advantage of nonparametric regression is its ability to highlight general trends in the data, such as those described earlier, and to attribute local fluctuations to unexplained random noise. The nonparametric regression model used by PROC GAM specifies that the underlying removal rates $\lambda _{j}$ are of the form

\[  \log \lambda _{j} = \beta _0 + \beta _1 \mi{Month}_ j + s(\mi{Month}_ j)  \]

where $\beta _1$ is a linear coefficient and $s(~ )$ is a nonparametric regression function. $\beta _1$ and $s(~ )$ define the linear and nonparametric parts, respectively, of the seasonal trend.

The following statements request that PROC GAM fit a cubic spline model to the monthly repair data. The output listing is displayed in Output 41.2.3 and Output 41.2.4.

title 'Analysis of Component Reliability';
title2 'Spline model';
proc gam data=equip;
   model removals=spline(month) / dist=Poisson method=gcv;

The METHOD=GCV option is used to determine an appropriate level of smoothing.

Output 41.2.3: PROC GAM Listing for Cubic Spline Regression Using the METHOD=GCV Option

Analysis of Component Reliability
Spline model

The GAM Procedure
Dependent Variable: removals
Smoothing Model Component(s): spline(month)

Summary of Input Data Set
Number of Observations 48
Number of Missing Observations 0
Distribution Poisson
Link Function Log

Output 41.2.4: Model Fit Statistics

Regression Model Analysis
Parameter Estimates
Parameter Parameter
t Value Pr > |t|
Intercept 1.34594 0.14509 9.28 <.0001
Linear(month) 0.02274 0.01893 1.20 0.2362

Smoothing Model Analysis
Fit Summary for Smoothing Components
Component Smoothing
Spline(month) 0.901512 1.879980 0.115848 12

Smoothing Model Analysis
Analysis of Deviance
Source DF Sum of Squares Chi-Square Pr > ChiSq
Spline(month) 1.87998 8.877764 8.8778 0.0103

Notice in the listing in Output 41.2.4 that the DF value chosen for the nonlinear portion of the spline by minimizing GCV is about 1.88, which is smaller than the default value of 3. This indicates that the spline model of the seasonal trend is relatively simple. As indicated by the "Analysis of Deviance" table, it is a significant feature of the data. The table lists a p-value of 0.0103 for the hypothesis of no seasonal trend. Note also that the "Parameter Estimates" table lists a p-value of 0.2362 for the hypothesis of no linear factor in the seasonal trend indicating no significant linear trend.

The following statements use ODS Graphics to plot the smoothing component for the effect of Month on predicted repair rates. The CLM suboption for the PLOTS=COMPONENTS option adds a 95% confidence band to the fit.

ods graphics on;

proc gam data=equip plots=components(clm);
   model removals=spline(month) / dist=Poisson method=gcv;

ods graphics off;

For general information about ODS graphics, see Chapter 21: Statistical Graphics Using ODS. For specific information about the graphics available in the GAM procedure, see the section ODS Graphics. The smoothing component plot is displayed in Output 41.2.5.

In Output 41.2.5, it is apparent that the pattern of repair rates follows the general pattern observed in Output 41.2.2. However, the plot in Output 41.2.5 is much cleaner because the month-to-month fluctuations are smoothed out to reveal the broader seasonal trend.

Output 41.2.5: Estimated Nonparametric Factor of Seasonal Trend, Along with 95% Confidence Bounds

Estimated Nonparametric Factor of Seasonal Trend, Along with 95% Confidence Bounds

In Output 41.2.1 the small p-value (p = 0.0321) for the hypothesis of no seasonal trend indicates that the data exhibit significant seasonal structure. Output 41.2.5 is a graphical illustration of the seasonality of the number of removals.