STATISTICS AND THE SOCIAL SCIENCES

New Statistical Modules for Marketing Research and Time-Series Analysis

by Robert A. Yaffee

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The ACF has recently acquired several new statistical modules for use with SPSS for Windows:

• The Trends module (discussed below) permits analysis of time series.
  
• The Exact Test module calculates exact probabilities for significance levels when the conventional statistical assumptions of normality and large sample size do not hold. When data are non-normally distributed, sparsely distributed, badly skewed, very unbalanced, or small in sample size, researchers would be able to use this module for proper estimation.
  
• Marketing researchers will find market- segmentation analysis possible with the new Chi-square Automatic Interaction Detection among categorical variables provided by the CHAID module.
  
• They may also find the new Correspondence and Conjoint analysis contained within the new Categories module useful.
  
• Both marketing researchers and forecasters may choose to experiment with market-data segmentation and forecasting, using the new neural-networks package called Neural Connections.
  

Time-Series Analysis

ARIMA and regression techniques are available. The ARIMA (autoregressive integrated moving average) procedure permits identification, estimation, diagnosis, metadiagnosis, and forecasting. For series afflicted with first-order autocorrelation, there is the autoregression procedure. For series afflicted with serious heteroskedasticity problems, there is the weighted least-squares regression procedure. For series with serious multicollinearity and simultaneity, there is the two-stage least-squares regression procedure, which may be used when proxy variables are available.

The Trends module under Windows produces textual and graphical output that can easily be incorporated in word-processing documents. The module has also been added to the SPSS under Unix on the ACF's IBM RS-6000/C-10 computer. A series of lectures on the use of the Trends is planned for the fall of 1996.

Neural Networks

Fundamentally, neural networks consist of basic components called neurons between which connections are formed. Functionally, the neurons are arrayed in three types of layers - input, hidden, and output. Three layers is a bare minimum for analyzing nonlinear problems, and there are often several hidden layers.

In Neural Connections, icons on the computer screen serve as manipulable symbols for input data, smoothing, filtering, modeling, or forecasting processes. The decision function may be depicted in a graph.

In the first phase of the iterative model-building cycle, the data are preprocessed. The neural net first transforms input and target variables to make them utilizable. Categorical and date-formatted variables are converted to numeric variables. Nonnormal variables are deskewed. Input variables are standardized, while outliers are clipped.

In the second phase, the neural net divides the data set into three subsets, one of which it will use for training, one for validation, and one for testing.

During the third phase, the neural net uses training data to form a system of connections between the layers of neurons for segmentation, classification, or prediction of the target (output) variable values.

As the network of connections is built between the layers, each connection is assigned a statistical weight. Input layers are stimulated by software algorithms to send signals to hidden layers. As a signal is sent through each connection, it is multiplied by the connection weight. Stimulated neurons receive signals from preceding layers of neurons and from a bias neuron. They sum these signals, subject this sum to a statistical transformation, and transmit the result to neurons in subsequent layers. The output of these neurons is a function of the net structure and the connection weights.

When the signaled output value is subtracted from the actual data value, the residual is an error. Errors are fed back from the output to the hidden layers for corrective adjustment of the connection weights to minimize the sum of squared errors. Training of the net takes place through this iterative feedforward of signal and feedback of the errors until a convergence of target with actual values takes place.

	A three-dimensional graph produced in SPSS's Neural Connections module represents decision output as a function of two inputs: age and maximum bank balance. The graph can be rotated as desired.

In the validation phase, the network is run with the validation data subset, in order to minimize overfitting the net to the peculiarity of the training sample. As the net is trained, errors of both training and validation decline. After a point, though, the validation error begins to increase; the optimal network configuration holds at the point of minimal validation error. After that point, any decrease in training error would indicate overtraining to the particular sample of data.

Multiple random starting points may be used to avoid local minimal error ruts and multiple data sets may be used for extended validation. When used for forecasting, the multiple starts are constrained by the time-dependent order of the data series. After fitting of the series, the forecast may be extrapolated from the fit of a model and projected into the future.

In these ways, the model may be cross-validated on other data sets and used for segmentation, classification, prediction, or forecasting.

The Neural Connections module allows for visualization of its decision criterion. A three-dimensional graph of the decisional surface of the neural net may be displayed in relation to other inputs as in the accompanying figure. In addition to this graph tool, another visualization device for sensitivity analysis shows how output would change in accordance with designed alterations of the graphed inputs. With the heuristic aid of these procedures, the neural net can be further refined or developed.

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Posted 15 February 1996. Revised 24 May 2004.