El paradigma de campo en T.G y la Nueva Psicología Social.

assumption validation. This is because the assumptions are about the error term. If the error term is not involved, the assumptions are not needed. However, if one wishes to do inference or model building, the assumptions must be verified.

INFERENCE IN REGRESSION

Consider the regression results given in Table 2.11. We have a predictor X and a responseY, and assume that we are unfamiliar with this type of data, except that each variable ranges from about−4 to 4. We are interested in usingX to predictY. Now the coefficient of determination takes on the valuer2_{=0.3%,which would tend to} indicate that the model is not at all useful. We are tempted to conclude that there is no linear relationship betweenxandy.

However, are we sure that there is no linear relationship between the variables? It is possible that such a relationship could exist even thoughr2 is small. The ques- tion is: Does there exist some systematic approach for determining whether a linear relationship exists between two variables? The answer, of course, is yes: Inference in regression offers a systematic framework for assessing the significance of linear association between two variables.

We shall examine four inferential methods in this chapter:

1. Thet-test for the relationship between the response variable and the predictor variable

2. The confidence interval for the slope,β1

3. The confidence interval for the mean of the response variable given a particular value of the predictor

4. The prediction interval for a random value of the response variable given a particular value of the predictor

In Chapter 3 we also investigate theF-test for the significance of the regression as a whole. However, for simple linear regression, thet-test and theF-test are equivalent. How do we go about performing inference in regression? Take a moment to consider the form of the regression equation:

y=β0+β1x+ε

TABLE 2.11 Regression That Is Not Very Useful, or Is It?

The regression equation is Y = 0.783 + 0.0559 X

Predictor Coef SE Coef T P

Constant 0.78262 0.03791 20.64 0.000

Y 0.05594 0.03056 1.83 0.067

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58 CHAPTER 2 REGRESSION MODELING

This equation asserts that there is a linear relationship between yon the one hand and some function ofx on the other. Now,β1is a model parameter, so that it is a constant whose value is unknown. Is there some value thatβ1could take such that ifβ1took that value, there would no longer exist a linear relationship betweenxand y?

Consider what would happen ifβ1 were zero. Then the regression equation would be

y=β0+(0)x+ε

In other words, whenβ1=0, the regression equation becomes y=β0+ε

That is, a linear relationship betweenxandyno longer exists. On the other hand, if β1takes on any conceivable value other than zero, a linear relationship of some kind exists between the response and the predictor. Much of our regression inference in this chapter is based on this key idea: that the linear relationship betweenx and y depends on the value ofβ1.

t-Test for the Relationship Betweenxandy

The least-squares estimate of the slope,b1, is a statistic. Like all statistics, it has a sampling distribution with a particular mean and standard error. The sampling distribution ofb1has as its mean the (unknown) value of the true slopeβ1and has as its standard error the following:

σb1 =

σ

x2−_x2_n

Just as one-sample inference about the mean is based on the sampling distribution of ¯

x,so regression inference about the slopeβ1is based on this sampling distribution ofb1.

The point estimate ofσb1issb1,given by sb1 =

s

x2−_x2_n

wheresis the standard error of the estimate, reported in the regression results. Thesb1 statistic is to be interpreted as a measure of the variability of the slope. Large values ofsb1 indicate that the estimate of the slopeb1is unstable, while small values ofsb1 indicate that the estimate of the slopeb1is precise.

Thet-test is based on the distribution of t =(β1−β1)/sb1, which follows a t-distribution withn – 2 degrees of freedom. When the null hypothesis is true, the test statistict=b1/sb1followsa t-distribution withn– 2 degrees of freedom.

To illustrate, we shall carry out the t-test using the results from Table 2.7, the regression of nutritional rating on sugar content. For convenience, part of Table 2.7 is reproduced here as Table 2.12. Consider the row in Table 2.12, labeled “Sugars. ”

INFERENCE IN REGRESSION 59

TABLE 2.12 Results of Regression of Nutritional Rating on Sugar Content

The regression equation is Rating = 59.4 - 2.42 Sugars

Predictor Coef SE Coef T P

Constant 59.444 1.951 30.47 0.000 Sugars -2.4193 0.2376 -10.18 0.000 S = 9.16160 R-Sq = 98.0% R-Sq(adj) = 57.5% Analysis of Variance Source DF SS MS F P Regression 1 8701.7 8701.7 103.67 0.000 Residual Error 75 6295.1 83.9 Total 76 14996.8

r _{Under “Coef” is found the value ofb1,−2.4193.}

r _{Under “SE Coef” is found the value ofsb1,the standard error of the slope. Here}

sb1=0.2376.

r _{Under “T” is found the value of thet-statistic, that is, the test statistic for the}

t-test,t =b1/sb1= −2.4193/0.2376= −10.18.

r _{Under “P” is found the p-value of the t-statistic. Since this is a two-tailed} test, thisp-value takes the following form:p-value=P(|t|>tobs),wheretobs represent the observed value of thet-statistic from the regression results. Here p-value=P(|t|>tobs)=P(|t|>−10.18)≈0.000,although, of course, no continuousp-value ever equals precisely zero.

The null hypothesis asserts that no linear relationship exists between the vari- ables, while the alternative hypothesis states that such a relationship does indeed exist.

r _{H0: β1=0 (There is no linear relationship between sugar content and nutri-} tional rating.)

r _{Ha: β1=0 (Yes, there is a linear relationship between sugar content and} nutritional rating.)

We shall carry out the hypothesis test using thep-value method, where the null hypothesis is rejected when thep-value of the test statistic is small. What determines how small is small depends on the field of study, the analyst, and domain experts although many analysts routinely use 0.05 as a threshold. Here, we have p-value ≈0.00,which is surely smaller than any reasonable threshold of significance. We therefore reject the null hypothesis and conclude that a linear relationship exists between sugar content and nutritional rating.

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