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Chapter 13: Problem 51

The article "The Undrained Strength of Some Thawed Permafrost Soils" (Canadian Geotechnical J., 1979: \(420-427\) ) contains the following data on undrained shear strength of sandy soil \((y\), in \(\mathrm{kPa})\), depth \(\left(x_{1}\right.\), in \(\left.\mathrm{m}\right)\), and water content \(\left(x_{2}\right.\), in \(\left.\%\right)\). a. Do plots of \(e^{*}\) versus \(x_{1}, e^{*}\) versus \(x_{2}\), and \(e^{*}\) versus \(\hat{y}\) suggest that the full quadratic model should be modified? Explain your answer. b. The value of \(R^{2}\) for the full quadratic model is .759. Test at level . 05 the null hypothesis stating that there is no linear relationship between the dependent variable and any of the five predictors. c. It can be shown that \(V(Y)=\sigma^{2}=V(\hat{Y})+V(Y-\hat{Y})\). The estimate of \(\sigma\) is \(\hat{\sigma}=s=6.99\) (from the full quadratic model). First obtain the estimated standard deviation of \(Y-\hat{Y}\), and then estimate the standard deviation of \(\hat{Y}\left(\right.\) i.e., \(\left.\hat{\beta}_{0}+\hat{\beta}_{1} x_{1}+\hat{\beta}_{2} x_{2}+\hat{\beta}_{3} x_{1}^{2}+\hat{\beta}_{4} x_{2}^{2}+\hat{\beta}_{3} x_{1} x_{2}\right)\) when \(x_{1}=8.0\) and \(x_{2}=33.1\). Finally, compute a \(95 \%\) CI for mean strength. [Hint: What is \(\left.(y-\hat{y}) / e^{*} ?\right]\) d. Fitting the first-order model with regression function \(\mu_{y_{\cdot} \cdot x_{2}}=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\) results in \(\mathrm{SSE}=894.95\), Test at level .05 the null hypothesis that states that all quadratic terms can be deleted from the model.

Short Answer

Expert verified
a. Assess scatter plots for residual patterns; b. Reject null: There's a linear relationship; c. Compute standard deviations, then CI for mean strength; d. Conduct F-test: Quadratic terms can be significant.

Step by step solution

01

Analyze Scatter Plots for Model Modification

In this step, observe scatter plots of residuals (\(e^*\)) against each predictor and the fitted values. If the residuals show a non-random pattern or a funnel shape, this suggests the model's misspecification and needs modifications. Compare the spread and trend of residuals across these plots to determine if improvements are needed.
02

Conduct Hypothesis Test for Linear Relationship

To test the null hypothesis that there is no linear dependency between the dependent variable and predictors, use an F-test. Calculate the F-statistic using the model's explained variance and the total variance. Compare it with the critical F-value at \(\alpha = 0.05\). If the F-statistic exceeds the critical value, reject the null hypothesis, indicating a significant linear relationship.
03

Calculate the Estimated Standard Deviation of Residuals

Use the given relation \(V(Y) = V(\hat{Y}) + V(Y-\hat{Y})\) to find \(V(Y-\hat{Y})\). With the provided standard error \(s = 6.99\) as \(V(Y)\), compute \(V(Y-\hat{Y})\) after finding \(V(\hat{Y})\). Subtract \(V(\hat{Y})\) from \(s^2\) to get \(V(Y-\hat{Y})\), then take its square root for the standard deviation.
04

Estimate the Standard Deviation of \(\hat{Y}\)

With estimates for \(x_1 = 8.0\) and \(x_2 = 33.1\), calculate \( \hat{Y} = \hat{\beta}_{0} + \hat{\beta}_{1} x_{1} + \hat{\beta}_{2} x_{2} + \hat{\beta}_{3} x_{1}^{2} + \hat{\beta}_{4} x_{2}^{2} + \hat{\beta}_{3} x_{1} x_{2} \). Estimate its standard deviation using the relation obtained in Step 3 and known statistics and coefficients of the model.
05

Compute 95% Confidence Interval for Mean Strength

Using the estimated standard deviation of \(\hat{Y}\), compute the margin of error for the confidence interval by multiplying the standard deviation by the critical value of the t-distribution (at \(95\%\) confidence and appropriate degrees of freedom). In the end, express the mean strength as \(\hat{Y} \pm\) margin of error.
06

Hypothesis Test for Quadratic Terms

Assess if \(SSE\) for the first-order model supports removing the quadratic terms by comparing it with the full model's SSE. Perform an F-test comparing the simplified model (without quadratic terms) against the full model. Calculate the F-statistic using differences in SSE divided by their corresponding degrees of freedom, and compare to critical values at \(\alpha = 0.05\). Reject the null hypothesis if the F-statistic suggests significant contributions of quadratic terms.

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Key Concepts

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

Regressions in Statistics
Regression analysis in statistics is a powerful tool used to understand the relationship between a dependent variable and one or more independent variables. In the context of undrained shear strength in thawed permafrost soils, regression helps in predicting the soil strength based on predictors such as depth and water content. The full quadratic model used here incorporates both linear and quadratic terms, representing not only the direct relationships but also the squared terms of predictors and their interactions.

Scatter plots of residuals are integral to diagnosing the fit of a regression model. If residuals (\(e^*\)) show patterns like funnels or systematic deviations, it often indicates a model that doesn't capture true variability. For example, a funnel shape suggests heteroscedasticity, where variance changes with levels of a predictor.

Improving a model might be necessary if residual patterns suggest misspecification. Adjusting models can involve removing insignificant predictors, transforming variables, or adding interaction terms. Quadratic terms, in this case, help model the curvilinear relationship between soil strength, depth, and water content, which might not be captured adequately by first-order (linear) terms alone.
  • Linear Regression: Involves straightforward relationships.
  • Quadratic Regression: Includes squared terms, accounting for curvature.
  • Residual Analysis: Checks model accuracy by examining prediction errors.
Hypothesis Testing in Engineering
In engineering, hypothesis testing is used to validate assumptions and determine the significance of model terms or interventions. For the undrained shear strength model, hypothesis tests ascertain the presence of meaningful relationships between predictors and the dependent variable.

The null hypothesis, in this case, might state that no linear relationship exists between the shear strength and any predictors like depth or water content. An F-test is a common approach to test this hypothesis. It compares the model's overall explanatory power versus the irrelevant model (one with no predictors).

If the computed F-value in this test surpasses a critical value at a specified significance level (e.g., \(\alpha = 0.05\)), the null hypothesis is rejected, suggesting that at least one predictor has a significant linear relationship with the dependent variable.
  • Null Hypothesis: Assumes no effect or relationship.
  • F-Test: Compares model variance against base variance to test significance.
  • Significance Level: Threshold (e.g., \(\alpha = 0.05\)) for deciding on hypothesis rejection.
Confidence Interval Estimation
Confidence intervals give an estimated range believed to contain the true mean of the dependent variable (e.g., undrained shear strength) at certain predictor values. Given assumptions hold, these intervals provide insight into the precision and reliability of the estimated parameter.

To compute a confidence interval for the mean strength, start by determining the estimated standard deviation associated with the predictor forecasts. This involves finding the variance of residuals and the predicted values using the model coefficients and observed data.

The margin of error is calculated by multiplying the standard deviation of the estimate by a critical value from the t-distribution (matching the confidence level, like \(95\%\)). Add and subtract this error from the point estimate to form an interval, representing where the mean likely lies within the given confidence.
  • Confidence Interval: A range surrounding the estimated parameter.
  • Margin of Error: Quantifies uncertainty in estimation.
  • T-distribution: Used when sample size is small or population standard deviation unknown.
Quadratic Model Evaluation
Evaluating a quadratic model involves assessing both its fit to the data and its complexity when compared to simpler models. The quadratic model is more complex than a simple linear model as it includes squared terms for predictors, capturing potential curvature in the relationship with the response variable, such as undrained shear strength.

Comparing quadratic and linear models can involve an SSE (Sum of Squared Errors) test. This test evaluates whether the quadratic terms significantly enhance the model's predictive ability. A higher SSE in linear models compared to quadratic ones suggests quadratic terms' importance, as they reduce unexplained variance.

The F-test can specifically compare linear and quadratic models by evaluating changes in explained variance. The test considers whether reductions in SSE due to adding quadratic terms are substantial compared to the increase in required parameters.
  • Model Complexity: Involves balancing fit and simplicity.
  • SSE: Measures how well the model captures data variability.
  • F-Test for Model Comparison: Evaluates significant differences between models.

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Most popular questions from this chapter

An aeronautical engineering student carried out an experiment to study how \(y=\) lift/drag ratio related to the variables \(x_{1}=\) position of a certain forward lifting surface relative to the main wing and \(x_{2}=\) tail placement relative to the main wing, obtaining the following data (Sratistics for Engineering Problem Solving, p. 133): \begin{tabular}{lcc} \(\boldsymbol{x}_{1}(\mathbf{i n .})\) & \(\boldsymbol{x}_{2}(\mathbf{i n} .)\) & \(\boldsymbol{y}\) \\ \hline\(-1.2\) & \(-1.2\) & \(.858\) \\ \(-1.2\) & 0 & \(3.156\) \\ \(-1.2\) & \(1.2\) & \(3.644\) \\ 0 & \(-1.2\) & \(4.281\) \\ 0 & 0 & \(3.481\) \\ 0 & \(1.2\) & \(3.918\) \\ \(1.2\) & \(-1.2\) & \(4.136\) \\ \(1.2\) & 0 & \(3.364\) \\ \(1.2\) & \(1.2\) & \(4.018\) \\ & & \(\bar{y}=3.428, \mathrm{SST}=8.55\) \end{tabular} a. Fitting the first-order model gives \(\mathrm{SSE}=5.18\), whereas including \(x_{3}=x_{1} x_{2}\) as a predictor results in \(\mathrm{SSE}=3.07\). Calculate and interpret the coefficient of multiple determination for each model. b. Carry out a test of model utility using \(\alpha=.05\) for each of the models described in part (a). Does either result surprise you?

A trucking company considered a multiple regression model for relating the dependent variable \(y=\) total daily travel time for one of its drivers (hours) to the predictors \(x_{1}=\) distance traveled (miles) and \(x_{2}=\) the number of deliveries made. Suppose that the model equation is $$ Y=-.800+.060 x_{1}+.900 x_{2}+\epsilon $$ a. What is the mean value of travel time when distance traveled is 50 miles and three deliveries are made? b. How would you interpret \(\beta_{1}=.060\), the coefficient of the predictor \(x_{1}\) ? What is the interpretation of \(\beta_{2}=.900\) ? c. If \(\sigma=.5\) hour, what is the probability that travel time will be at most 6 hours when three deliveries are made and the distance traveled is 50 miles?

Let \(y=\) sales at a fast-food outlet \((1000 \mathrm{~s}\) of \(\$), x_{1}=\) number of competing outlets within a 1-mile radius, \(x_{2}=\) population within a 1-mile radius ( \(1000 \mathrm{~s}\) of people), and \(x_{3}\) be an indicator variable that equals 1 if the outlet has a drive-up window and 0 otherwise. Suppose that the true regression model is $$ Y=10.00-1.2 x_{1}+6.8 x_{2}+15.3 x_{3}+\epsilon $$ a. What is the mean value of sales when the number of competing outlets is 2 , there are 8000 people within a 1-mile radius, and the outlet has a drive-up window? b. What is the mean value of sales for an outlet without a drive-up window that has three competing outlets and 5000 people within a 1 -mile radius? c. Interpret \(\beta_{3}\).

An experiment carried out to study the effect of the mole contents of cobalt \(\left(x_{1}\right)\) and the calcination temperature \(\left(x_{2}\right)\) on the surface area of an iron-cobalt hydroxide catalyst \((y)\) resulted in the accompanying data ("Structural Changes and Surface Properties of \(\mathrm{Co}_{x} \mathrm{Fe}_{3-\mathrm{x}} \mathrm{O}_{4}\) Spinels," \(J\) of Chemical Tech. and Biotech., 1994: 161-170). A request to the SAS package to fit \(\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}\), where \(x_{3}=x_{1} x_{2}\) (an interaction predictor) yielded the output below. b. Since \(\hat{\beta}_{1}=-46.0\), is it legitimate to conclude that if cobalt content increases by 1 unit while the values of the other predictors remain fixed, surface area can be expected to decrease by roughly 46 units? Explain your reasoning. c. Does there appear to be a useful linear relationship between \(y\) and the predictors? d. Given that mole contents and calcination temperature remain in the model, does the interaction predictor \(x_{3}\) provide useful information about \(y\) ? State and test the appropriate hypotheses using a significance level of \(.01\). e. The estimated standard deviation of \(Y\) when mole contents is \(2.0\) and calcination temperature is 500 is \(s_{\hat{\gamma}}=4.69\). Calculate a \(95 \%\) confidence interval for the mean value of surface area under these circumstances.

Feature recognition from surface models of complicated parts is becoming increasingly important in the development of efficient computer-aided design (CAD) systems. The article "A Computationally Efficient Approach to Feature Abstraction in Design-Manufacturing Integration" (J. of Engr: for Industry, 1995: 16-27) contained a graph of logadtotal recognition time), with time in sec, versus \(\log _{10}\) (number of edges of a part), from which the following representative values were read: \(\begin{array}{lrrrrrr}\text { Log(edges) } & 1.1 & 1.5 & 1.7 & 1.9 & 2.0 & 2.1 \\ \text { Log(time) } & .30 & .50 & .55 & .52 & .85 & .98 \\ \text { Log(edges) } & 2.2 & 2.3 & 2.7 & 2.8 & 3.0 & 3.3 \\ \text { Log(time) } & 1.10 & 1.00 & 1.18 & 1.45 & 1.65 & 1.84 \\ \text { Log(edges) } & 3.5 & 3.8 & 4.2 & 4.3 & & \\ \text { Log(time) } & 2.05 & 2.46 & 2.50 & 2.76 & & \end{array}\) a. Does a scatter plot of \(\log (\) time \()\) versus \(\log (\) edges) suggest an approximate linear relationship between these two variables? b. What probabilistic model for relating \(y=\) recognition time to \(x=\) number of edges is implied by the simple linear regression relationship between the transformed variables? c. Summary quantities calculated from the data are $$ \begin{aligned} &n=16 \quad \Sigma x_{i}^{\prime}=42.4 \quad \Sigma y_{i}^{\prime}=21.69 \\ &\Sigma\left(x_{i}^{\prime}\right)^{2}=126.34 \quad \Sigma\left(y_{i}^{\prime}\right)^{2}=38.5305 \\ &\Sigma x_{i}^{\prime} y_{i}^{\prime}=68.640 \end{aligned} $$ Calculate estimates of the parameters for the model in part (b), and then obtain a point prediction of time when the number of edges is 300 .
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