Question

The article “Readability of Liquid Crystal Displays: A Response Surface" (Human Factors [1983]: $185-190$ ) used an estimated regression equation to describe the relationship between $y=$ error percentage for subjects reading a four-digit liquid crystal display and the independent variables $x_1=$ level of backlight, $x_2=$ character subtense, $x_3=$ viewing angle, and $x_4=$ level of ambient light. From a table given in the article, SSRegr $=19.2,$ SSResid $=20.0$, and $n=30$. a. Does the estimated regression equation specify a useful relationship between $y$ and the independent variables? Use the model utility test with a .05 significance level. b. Calculate $R^2$ and $s_e$ for this model. Interpret these values. c. Do you think that the estimated regression equation would provide reasonably accurate predictions of error percentage? Explain.

Short answer

a. The calculated F-statistic is 2.4. Depending on the critical value from the F-distribution table, it can be determined if the model shows a useful relationship. b. The $R^2$ value is 0.49, indicating a moderate fit and the standard error $s_e$ is 0.482 which suggests a fairly accurate model. c. Given these values, it can be said that the model is likely to provide reasonably accurate predictions for the error percentage.

Step-by-step solution

Use Model utility test

A model utility test is performed to check if the regression equation is useful in determining the relationship between the independent and dependent variables. This can be done using an F-test, with a .05 significance level. The formula to calculate the F-statistic is $F=\frac{S{S}_{Regr}/k}{S{S}_{Resid}/(n-k-1)}$, where $k$ is the number of independent variables, $n$ is the total number of observations. In this problem, $S{S}_{Regr}=19.2$, $S{S}_{Resid}=20.0$, $k=4$, and $n=30$. So, calculating it yields $F=2.4$. In order to reject the null hypothesis—that there's no relationship between the variables—the calculated F statistic needs to be greater than the critical F-value from the F-distribution table with degrees of freedom $df1=k$ and $df2=n-k-1$. Comparing the calculated F-value with the critical F-value from the F-table will allow us to determine if the regression equation provides a useful relationship between $y$ and the independent variables.

Calculate and interpret $R^2$

The coefficient of determination $R^2$ can be calculated using the formula $R^2=S{S}_{Regr}/(S{S}_{Regr}+S{S}_{Resid})$. This value can range from 0 to 1, where a higher value indicates a better fit of the model to the data. The closer the value of $R^2$ is to 1, the better the model is at explaining the total variance in the data. Substituting the given values, we find $R^2=0.49$. This means that 49% of the variability in the dependent variable can be explained by the regression model, which is moderately good, as the explanation rate is close to 50%.

Calculate the standard error of the estimate and interpret it

The standard error of the estimate ($s_e$) gives an indication of the typical error in predicting $y$. It can be calculated using the formula $s_e=\sqrt{S{S}_{Resid}/(n-k-1)}$. This value gives the spread of the residuals—smaller values indicate a more accurate model. On calculation, $s_e$ comes out to be $0.482$. This value implies that there will be on average a 0.482 percentage point error when predicting error percentage using this model.

Reasoning the accuracy of the model

While the $R^2$ value is just under 50%, which indicates that the model does a fair job of explaining the data, the standard error of the estimate is also fairly small. Therefore, while the model so far might not be perfect, it could probably provide reasonable predictions for the error percentage. More factors could be taken into account to refine the model further based on domain knowledge and statistical significance.

Key concepts

Error Percentage Prediction

Predicting error percentages involves analyzing the deviations between actual observed values and the values predicted by the regression model. In this context, a regression model is constructed to relate the error percentage in reading a liquid crystal display to several independent variables like backlight level and ambient light. The predictive accuracy of this model is crucial for ensuring that it can reliably anticipate situations affecting readability.

To evaluate the model's accuracy, we look at parameters such as the coefficient of determination and the standard error, which provide insights into the model's effectiveness. The closer the predictions match actual observations, the more reliable the model is deemed to be for practical use. In any prediction model, assessing whether it can provide accurate outcomes is fundamental, especially when the implications affect usability and performance environments.

Model Utility Test

The model utility test is a statistical method used to determine if a regression model offers a meaningful association between independent variables and a dependent variable. This involves performing an F-test. This test assesses whether the regression model reduces uncertainty in predicting the dependent variable better than a model without any predictors, essentially questioning if the added complexity of predictors is justified.

The test employs the F-statistic formula: $$F=\frac{{\text{SS}}_{Regr}/k}{{\text{SS}}_{Resid}/(n-k-1)}$$ This compares explained variance to unexplained variance. We calculate this statistic and compare it to a critical F-value from an F-distribution table. If the computed F-value exceeds the critical value, the model is considered useful. This procedure helps researchers decide whether to retain or modify their predictive models.

Coefficient of Determination

The coefficient of determination, denoted as $R^2$, is a metric that quantifies how well a regression model explains the variability of the dependent variable. It is defined by the formula: $$R^2=\frac{{\text{SS}}_{Regr}}{{\text{SS}}_{Regr}+{\text{SS}}_{Resid}}$$The result ranges from 0 to 1, where a value closer to 1 indicates that the model explains a large portion of the variance. In simpler terms, it shows the strength of the relationship between the model and the dependent variable.

For instance, an $R^2$ value of 0.49 suggests that approximately 49% of the variance in error percentage is captured by the model, indicating a moderate level of accuracy. Researchers use this value to judge a model's performance and to decide if further enhancements or additional variables are needed.

Standard Error of Estimate

The standard error of estimate, represented as $s_e$, provides a measure of how much observed values deviate from the predicted values set by the regression model. It is akin to the standard deviation but in the context of a regression line.

This statistic is calculated using the formula: $$s_e=\sqrt{\frac{{\text{SS}}_{Resid}}{n-k-1}}$$ Where a smaller $s_e$ indicates tighter clusters of data points around the estimated regression line, implying more accurate predictions. In the scenario provided, $s_e$ calculated at 0.482 means that the model's predictions will deviate, on average, by around 0.482 percentage points from actual observations. Evaluating $s_e$ helps in understanding model precision and assessing where potential improvements can be made to enhance forecast accuracy.