Probability Approximations and Beyond: 205 (Lecture Notes in Statistics)

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Montgomery Give your final model. Give p. Fit this model and estimate the parameters. Special Nonlinear Models for Nonideal Conditions In much of the preceding material in this chapter and in C h a p t e r 11 wc have benefited substantially from the assumption t h a t the model errors, the ei, are normal with mean zero and constant variance o. However, there are many reallife situations in which the response is clearly nonnormal. For example, a wealth of applications exist where the r e s p o n s e is b i n a r y 0 or 1 and hence Bernoulli in n a t u r e.

In the1 social sciences the problem may be to develop a model to predict whether or not an individual is a good credit risk or not 0 or 1 as a function of certain socioeconomic regressors such as income, ago, gender and level of education.

In a biomedical drug trial the response is often whether or not the patient responds positively to a drug while regressors may include drug dosage as well as biological factors such us age, weight, and blood pressure. Again the response is binary. Applications are also abundant in manufacturing areas where certain controllable factors influence whether a manufactured item is defective or not. A second type of nonnormal application on which we will touch briefly has to do with count data.

Here the assumption of a Poisson response is often convenient. In biomedical applications the number of cancer cell colonies may be the response which is modeled against drug dosages. In the textile industry the number of imperfections per yard of cloth may be a reasonable response which is modeled against certain process variables.

Nonhomogeneous Variance The reader should note the comparison of the ideal i. We have become accustomed to the fact that the normal case is very special in that the variance is independent of the mean. Clearly this is not the case for either Bernoulli or Poisson responses. For example, if the response is 0 or 1, suggesting a Bernoulli response, then the model is of the form.

1. Introduction to Statistics

As a result, the variance is not constant. This rules out the use of standard least squares that we have utilized in our linear regression work up to this point. The same is true for the Poisson case since the model is of the form. Binary Response Logistic Regression The most popular approach to modeling binary responses is a technique entitled logistic regression. It is used extensively in the biological sciences, biomedical research, and engineering. Indeed, even in the social sciences binary responses are found to be plentiful. The basic distribution for the response is either Bernoulli or binomial.

The former is found in observational studies where there are no repeated runs at each regressor level while the latter will be the case when an experiment is designed. For example, in a clinical trial in which a new drug is being evaluated the goal might be to determine the dose of the drug that provides efficacy.

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So certain doses will be employed in the experiment and more than one subject will be used for each dose. This case is called the grouped case. W h a t Is t h e Model for Logistic Regression? In the case of binary responses the mean response is a probability. In the preceding clinical trial illustration, we might say that we wish to estimate the probability that.

Thus the model is written in terms of a probability.

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Degrees of freedom (statistics)

Of course, we do not rule out involving multiple regressors and polynomial terms in the so-called linear predictor. Characteristics of Logistic Function A plot of the logistic function reveals a great deal about its characteristics and why it is utilized for this type of problem.


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First, the function is nonlinear. In addition, the plot in Figure Thus we would never experience an estimated probability exceeding 1. The regression coefficients in the linear predictor can be estimated by the method of maximum likelihood as described in Chapter 9. The solution to the likelihood equations involves an iterative methodology that will not be described here.

However, we will present an example and discuss the computer printout and conclusions. The results show the effect of different doses of nicotine on the common fruit fly. The purpose of the experiment was to use logistic regression to arrive at an appropriate model relating probability of "kill" to concentration.

In addition, the. Of particular interest is the ED50, the concentration that produces a 0. E s t i m a t e of Effective Dose The estimate of ED50 is found very simply from the estimate 60 for 0Q and 61 for 0x. From the logistic function, we see that. Concept of Odds Ratio A n o t h e r form of inference t h a t is conveniently accomplished using logistic regression is derived from the use of the odds ratio.

For example, in the case of E x a m p l e Definition 1 2. In logistic regression an odds ratio is the ratio of odds of success at condition 2 to that of condition 1 in the regressort , that is,. This allows the analyst to ascertain a sense of the utility of changing t h e regressor by a certain number of units. T h e implication of an odds ratio of 3. Review Exercises The regressor variables are as follows: average depth of 50 cells xi : area of in-stream cover i. The response is y, the fish biomass.

The data are as follows: Obs. Compute these statistics for all possible subsets. Two levels of each variable were chosen and measurements corresponding to the coded independent variables were recorded as follows: y. Compute the prediction criteria for the reduced model. Comment on the appropriateness of x4 for prediction of the heat transfer coefficient.

Statistical mechanics: the Riemann zeta function interpreted as a partition function

Comment of the results. Quote the final model. Make a decision and quote the final model. Suppose it is of interest to add some "interaction" terms.

Section 31.200

Why or why not? T3 removed. Give P-value and conclusions. The CO2 floods oil pockets and displaces the crude oil. In the experiment, flow tubes are dipped into sample oil podcets containing a known amount of oil. Using three different values of flow pressure and three different values of dipping angles the oil pockets are flooded with CO2, and the percentage of oil displaced recorded. Fit the model above to the data, and suggest any model editing that may be needed.

Also measured are the dispersion a;i, and dipolar and hydrogen bonding solubility parameters as, and A portion of the data is shown in the table below. In the model, y is the negative logarithm of the mole fraction. The factors considered were a healthy diet with an exercise program, the typical dosage of medication for hypertension, and no intervention. The pretreatment body mass index BMI was also calculated because it is known to affect blood pressure. The response considered in this study.

The variable group has the following levels. Doess it appear that exercise and diet could be effectively useid to lower blood pressure? Explain your answer from the results. You may wish to form the model in more than one way to answer both of these questions. The data set is repeated here. Goals are to do outlier detection and eventually determine which model terms are to be used in the final model.

1. Introduction

Residual 46 Std Error Residual Five dosage levels were applied to the rabbits used for the experiment. The data are. See Myers, , in the bibliography. The numbers of ''failures" were observed. The data are as follows: N u m b e r of N u m b e r of Load Specimens Failures 13 5 95 35 70 80 95 90 a Use: logistic regression to fit the model 1!

Potential Misconceptions and Hazards; Relationship to Material in Other Chapters There arc several procedures discussed in this chapter for use in the " a t t e m p t " to find the best model. However, one of the: most important misconceptions under which naive scientists or engineers labor is t h a t there is a t r u e linear m o d e l and that it, can be found.

In most scientific phenomena, relationships between scientific variables are nonlinear in nature and the true: model is unknown. At times the choice of the model to be adopted may depend on what information needs to be derived from the model.


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  • Is it to be used for prediction? Is it to be used for the purpose of explaining the role of each regressor? This ''choice" can be made difficult in the presence of collincarity. It is true t h a t for many regression problems there are multiple models t h a t arc very similar in performance.

    See the Myers reference for details. One of the most damaging misuses of the material in this chapter is to apply too much importance to R 2 in the choice of the so-called best model. Much attention is given in this chapter to outlier detection. A classical serious misuse of statistics may center around the decision made concerning the detection of outliers.

    We hope it, is clear that the analyst should absolutely not carry out the exercise of detecting outliers, eliminate them from the d a t a set, fit a new model, report outlier detection, and so on. This is a tempting and disastrous procedure for arriving at a model that fits the d a t a well, with the result, being an example.