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07 - 11 - 2014
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## Curve estimation

Curve estimation procedure allows quickly determining character of relationship between variables. It is especially useful when relationship is not necessary linear.

Let us assess relationship between optical density and glucose content from the data given in Example 6. Curve estimation procedure proposes the choice of 11 models, however, in order to test only the most appropriate models, at beginning we may visually assess dependence between variables on scatterplot.

To build scatterplot we may select Chart Builder from Graphs menu, as it was described above in the correlation analysis.

Another variant:

1. Click the Graphs menu, point to Legacy Dialogs and select the Scatter/Dot…. diagram:

The Scatter/Dot dialog box opens:

2. Select the type of scatterplot (Simple Scatter) and click the Define button. The Simple Scatterplot dialog box opens:

3. Select the variable “Glucose” for X axis, click the respective transfer button ; the variable is moved to the X Axis: list box; then select the variable “OD” for Y axes in the same way.

4. Click the OK button. Scatterplot will be built in the Output Viewer window:

From the resulting scatterplot it is seen that relationship between variables is the most close to linear or quadratic, therefore, for the curve estimation we may assess only these models.

To specify the curve estimation procedure:

1. Click the Analyze menu, point toRegression and select Curve Estimation… :

The Curve Estimation dialog box opens:

2. Select the dependent variable (“OD”), click the upper transfer button , the variable is moved to the Dependent(s): list box. Select the independent variable (“Glucose”), click the respective transfer button , the variable is moved to the Independent list box.

3. By default only linear model is selected. Select also the Quadratic check box.

4. Click the OK button. The results will be displayed in the Output Viewer window.

The results of the curve estimation procedure include four tables - Model Description (lists variables in the model), Case Processing Summary (shows number of total, excluded, forecasted and newly created cases), Variable Processing Summary (lists number of positive values, zeros, negative and missing values) and the most important table – Model Summary and Parameter Estimates:

If ANOVA table was not requested in the Curve Estimation main dialog box, summary for all tested models are shown in a single Model Summary and Parameter Estimates table. For both, linear and quadratic, models significance is less than 0.05, therefore, the variation explained by each model is not due to chance.

In the column Parameter Estimates the resulting models are present as following.

Linear model: OD = -0.011+ 1.014×Glucose.

Quadratic model: OD = -0.457 + 2.506×Glucose – 1.127×(Glucose)2.

From the linear model we see that optical density changes almost the same as glucose concentration. However, the second negative coefficient in the quadratic model indicates that after some limit, an increase in glucose concentration will influence adversely optical density (concentration which exceeds optimal will not cause further stimulation of the growth of S. aureus). More exactly, increasing glucose concentration past 2.506/(2×1.127) = 1.11 will decrease the OD, as it is predicted by the model.

R square coefficient assesses the strength of relationship between the observed and model-predicted values of the dependent variable, and is the most important in assessing which model is better. For both models, values of R square coefficient are large, but for quadratic model it is larger, which indicates its better fitting with the data.

Along with the tables, the Output Viewer window contains the Curve fit chart, where we also can see that quadratic model better follows the shape of the data: