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Chapter 14: Problem 29

The article "Effect of Manual Defoliation on Pole Bean Yield" (Journal of Economic Entomology [1984]: \(1019-1023\) ) used a quadratic regression model to describe the relationship between \(y=\) yield \((\mathrm{kg} /\) plot \()\) and \(x=\mathrm{de}-\) foliation level (a proportion between 0 and 1 ). The estimated regression equation based on \(n=24\) was \(\hat{y}=\) \(12.39+6.67 x_{1}-15.25 x_{2}\) where \(x_{1}=x\) and \(x_{2}=x^{2}\). The article also reported that \(R^{2}\) for this model was. 902 . Does the quadratic model specify a useful relationship between \(y\) and \(x\) ? Carry out the appropriate test using a \(.01\) level of significance.

Short Answer

Expert verified
The short answer will depend on the outcome of the F-statistic and the comparison to the F critical value for \( df_{model} = 2 \) and \( df_{residual} = 21 \) at \( 伪 = 0.01 \). If the F-statistic > F critical value, we reject the null hypothesis and conclude that the regression model is useful. If the F-statistic < F critical value, we fail to reject the null hypothesis and cannot conclude that the model is useful.

Step by step solution

01

Identify the Hypotheses

The null hypothesis (\(H_{0}\)) is that the model isn鈥檛 useful for predicting the outcome (i.e., the model does not fit the data). Appropriately, this means that the coefficients for \(x_{1}\) (defoliation level) and \(x_{2}\) (defoliation level squared) in our regression model equal zero. Thus, \(H_{0}: 脽_{1} = 脽_{2} = 0\). The alternate hypothesis (\(H_{a}\)), however, posits that the model is useful in predicting the outcome, i.e., at least one of the coefficients is non-zero: \(H_{a}: \) at least one of \(脽_{1}, 脽_{2} \neq 0\).
02

Calculate the F statistic

The F statistic for the ANOVA test in a regression model is calculated by the formula: \[ F = \frac{((SSR/df_{model})/(SSE/df_{residual}))}{} \]. Here, 'SSR' refers to the sum of squares due to regression (the sum of squared distances between the predicted and mean values), and 'SSE' refers to the sum of squares due to error (the sum of squared distances between the actual and predicted values). 'df' stands for degrees of freedom. In general, for a model with 'p' predictor variables, the degrees of freedom for the model is 'p', while the degrees of freedom for the residual is 'n - p - 1', where 'n' is the sample size. For the model in question, the given information \(n=24\) and \(R^2 = 0.902\) are used to determine SSR, SSE, and the degrees of freedom.
03

Calculate the F Critical Value and Check against F statistic

First, it is important to use the F-distribution tables or a statistical software program with two degrees of freedom for the model (for the two predictor variables \(x_{1}\) and \(x_{2}\)) and 21 degrees of freedom for the residuals (n - p - 1 = 24 - 2 - 1 = 21) to find the critical value of F for a two-tailed test with alpha = 0.01. If the calculated F-statistic exceeds this critical value, the null hypothesis can be rejected.
04

Make the Decision

If the test statistic is greater than the critical value from the F distribution tables, then reject the null hypothesis, stating that the model is useful in predicting the yield. If not, fail to reject the null hypothesis, and conclude that there is not enough evidence to suggest that the model is useful in predicting the yield.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing
In the context of quadratic regression analysis, hypothesis testing is used to assess whether the model provides a reliable prediction of the outcome. Here, the objective is to determine if the model's coefficients contribute significantly to predicting the yield based on defoliation levels.

Hypothesis testing begins with setting up two competing hypotheses: the null hypothesis ( ) and the alternative hypothesis ( ).
  • Null Hypothesis ( ): Suggests that the model is not useful, which implies both coefficients in our regression equation are zero ( 鈥�: ).
  • Alternative Hypothesis ( ): At least one coefficient is not zero ( : at least one of , is not zero), indicating a useful model.
The next steps involve statistical tests to verify these hypotheses using a specified significance level, which in this exercise is 0.01. The critical step is comparing the calculated F-statistic with the critical value, derived from an F-distribution table, to make an informed conclusion about the hypotheses.
F-statistic Computation
Once the hypotheses are established, computing the F-statistic is essential to evaluate whether the model is significantly better than a model with no predictor variables at the specified level of significance.

The F-statistic is calculated using the formula: , where:
  • SSR (Sum of Squares due to Regression) indicates variation explained by the model.
  • SSE (Sum of Squares due to Error) indicates variation not explained by the model.
  • and are degrees of freedom for the model and residuals respectively.
For this particular analysis, with two predictor variables and a sample size ( =24), we determine:
  • Model degrees of freedom = 2 (for predictors and ).
  • Residual degrees of freedom = - -1 = 21.
By comparing the F-statistic to the critical value from F-distribution tables (specific to our degrees of freedom), we can decide whether to reject the null hypothesis.
R-squared Interpretation
In regression analysis, the R-squared ( ) value is crucial for understanding how well the model explains variability in the response variable. For the given model, an value of 0.902 implies that approximately 90.2% of the variation in yield is accounted for by the model incorporating defoliation level and its square.

This high value indicates a strong relationship between the defoliation levels and yield, signifying that the quadratic model fits the data well.
  • High : Suggests a good fit, with less unexplained variability.
  • However, it's essential to note that a high does not validate the model's plausibility or causation; it merely indicates a strong correlation.
Thus, while an of 0.902 is an encouraging sign of the model鈥檚 effectiveness, it should be interpreted along with hypothesis testing and F-statistic results to comprehensively assess model usefulness.

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Most popular questions from this chapter

The article "Readability of Liquid Crystal Displays: A Response Surface" (Human Factors \([1983]: 185-190)\) used the 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 rate? Explain.

The article "The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carburetted Water-Gas" (Society of Chemical Industry Joumal \([1946]: 166-168\) ) presented data on \(y=\) tar content (grains/100 \(\mathrm{ft}^{3}\) ) of a gas stream as a function of \(x_{1}=\) rotor speed \((\mathrm{rev} / \mathrm{min})\) and \(x_{2}=\) gas inlet temperature ( F). A regression model using \(x_{1}, x_{2}, x_{3}=x_{2}^{2}\) and \(x_{4}=x_{1} x_{2}\) was suggested: $$ \begin{aligned} \text { mean } y \text { value }=& 86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} \\ &+.001 x_{4} \end{aligned} $$ a. According to this model, what is the mean \(y\) value if \(x_{1}=3200\) and \(x_{2}=57 ?\) b. For this particular model, does it make sense to interpret the value of any individual \(\beta_{i}\left(\beta_{1}, \beta_{2}, \beta_{3}\right.\), or \(\left.\beta_{4}\right)\) in the way we have previously suggested? Explain.

A manufacturer of wood stoves collected data on \(y=\) particulate matter concentration and \(x_{1}=\) flue temperature for three different air intake settings (low, medium, and high). a. Write a model equation that includes dummy variables to incorporate intake setting, and interpret all the \(\beta\) coefficients. b. What additional predictors would be needed to incorporate interaction between temperature and intake setting?

If we knew the width and height of cylindrical tin cans of food, could we predict the volume of these cans with precision and accuracy? a. Give the equation that would allow us to make such predictions. b. Is the relationship between volume and its predictors, height and width, a linear one? c. Should we use an additive multiple regression model to predict a volume of a can from its height and width? Explain. d. If you were to take logarithms of each side of the equation in Part (a), would the relationship be linear?

Explain the difference between a deterministic and a probabilistic model. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) deterministically. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) in a probabilistic fashion.
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