Question

The article "Readability of Liquid Crystal Displays: A Response Surface" (Human Factors \([1983]: 185-190\) ) used a multiple regression model with four independent variables, where \(y=\) error percentage for subjects reading a four-digit liquid crystal display $$ \begin{aligned} &\left.x_{1}=\text { level of backlight (from } 0 \text { to } 122 \mathrm{~cd} / \mathrm{m}\right) \\ &x_{2}=\text { character subtense }\left(\text { from } .025^{\circ} \text { to } 1.34^{\circ}\right) \end{aligned} $$ \(x_{3}=\) viewing angle \(\left(\right.\) from \(0^{\circ}\) to \(60^{\circ}\) ) \(x_{4}=\) level of ambient light (from 20 to \(1500 \mathrm{~lx}\) ) The model equation suggested in the article is $$ y=1.52+.02 x_{1}-1.40 x_{2}+.02 x_{3}-.0006 x_{4}+e $$ a. Assume that this is the correct equation. What is the mean value of \(y\) when \(x_{1}=10, x_{2}=.5, x_{3}=50\), and \(x_{4}=100 ?\) b. What mean error percentage is associated with a backlight level of 20 , character subtense of \(.5\), viewing angle of 10, and ambient light level of 30 ? c. Interpret the values of \(\beta_{2}\) and \(\beta_{3}\)

Short answer

The mean value of y when \(x_{1}=10, x_{2}=.5, x_{3}=50\), and \(x_{4}=100 \) is 2.74. The mean error percentage when \(x_{1}=20, x_{2}=.5, x_{3}=10\), and \(x_{4}=30 \) is 1.92. The interpretation of the regression coefficients \(\beta_{2}\) and \(\beta_{3}\) is that for every unit increase in character subtense, the error percentage decreases by 1.4. If viewing angle is increased by one unit, the error percentage would increase by .02, all other factors remaining constant.

Step-by-step solution

Calculate mean value of y for given parameters

Given the equation \(y=1.52+.02 x_{1}-1.40 x_{2}+.02 x_{3}-.0006 x_{4}+e\), replace each variable with the given values \(x_{1}=10, x_{2}=.5, x_{3}=50\), and \(x_{4}=100 \). Then compute the result.

Calculate mean error percentage for given parameters

Using the same equation, substitute each variable with the given values \(x_{1}=20, x_{2}=.5, x_{3}=10\), and \(x_{4}=30 \). Then compute the result.

Interpret the values of Beta coefficients

Given the coefficient \(\beta_{2} = -1.40\), this means that for every unit increase in character subtense, the error percentage decreases by 1.4, all other factors being equal. The coefficient \(\beta_{3} = +.02\) signifies that for every unit increase in viewing angle, the error percentage increases by .02, all other factors being equal.

Key concepts

Independent Variables

When dealing with a multiple regression model, it's crucial to understand the role of independent variables. These are the input factors or predictors that researchers believe will have an impact on the dependent variable—in this case, the error percentage for subjects reading a liquid crystal display.

In our exercise, we have four independent variables: the level of backlight, character subtense, viewing angle, and level of ambient light. Each of these variables potentially influences how easily the display can be read, which is quantitatively expressed by the error percentage.

Technically, when statisticians apply a multiple regression analysis, they are investigating how each independent variable impacts the dependent variable while controlling for the effects of all other independent variables in the model. This helps to isolate the effect of each predictor on the outcome, which is a powerful tool for understanding complex relationships.

Error Percentage

Error percentage is a crucial measure in many research scenarios, particularly in human factors engineering as in our exercise. It refers to the rate at which errors occur when subjects perform a specific task, such as reading numbers on a liquid crystal display.

In multiple regression models, error percentage may serve as a dependent variable that the model attempts to predict. This dependent variable is what the equation outputs after accounting for the influence of each independent variable. It's also essential to note that the 'e' in our regression equation represents the random error term, which captures the variation in error percentage not explained by our model.

Practically speaking, when you're analyzing the results of such a model, a lower error percentage would suggest a better readability of the display. This, in turn, has direct implications for design decisions, like adjusting the level of backlight or ambient light to optimize display visibility and user performance.

Data Analysis

Data analysis is the backbone of understanding and leveraging the multiple regression model. Through this process, researchers can make sense of the collected data, applying statistical methods to conceive meaningful patterns or relationships.

In context with our exercise, the data analysis involves using the regression equation to calculate the mean value of the dependent variable (error percentage) for given sets of independent variables. This allows for the prediction and interpretation of how changes in environmental and display conditions affect the readability of liquid crystal displays.

For instance, by analyzing and interpreting the data, one might conclude that increasing the viewing angle will marginally increase the error percentage, whereas the size of the character (character subtense) has a considerably more substantial effect on reducing error percentage. Such insights are invaluable in the process of optimizing product design for user efficiency and satisfaction.