Predictor Cutoff

Examples of Predictor Cutoff in the following topics:

• Employee Selection

  • Two major factors determining the quality of a newly hired employee are predictor validity and selection ratio.
  • The predictor cutoff is a limit distinguishing between passing and failing scores on a selection test—people with scores above it are hired or further considered while those with scores below it are not.
  • This cutoff can be a very useful hiring tool, but it is only valuable if it is actually predictive of the type of performance the hiring managers are seeking.
  • SAT scores used as university admissions criteria are a good example of the use of predictor cutoff.

• Modeling the probability of an event

  • It is the probability pi that we model in relation to the predictor variables.
  • In our spam example, there are 10 predictor variables, so k = 10.
  • We used statistical software to fit the logistic regression model with all ten predictors described in Table 8.13.
  • Using backwards elimination with a p-value cutoff of 0.05 (start with the full model and trim the predictors with p-values greater than 0.05), we ultimately eliminate the exclaim subj, dollar, inherit, and cc predictors.
  • This is usually due to colinearity in the predictor variables.

• The normal approximation breaks down on small intervals

  • The normal approximation to the binomial distribution for intervals of values is usually improved if cutoff values are modified slightly.The cutoff values for the lower end of a shaded region should be reduced by 0.5, and the cutoff value for the upper end should be increased by 0.5.

• Including and assessing many variables in a model

  • Sometimes there are underlying structures or relationships between predictor variables.
  • A multiple regression model is a linear model with many predictors.
  • when there are k predictors.
  • If we examined the data carefully, we would see that some predictors are correlated.
  • Example 8.8 describes a common issue in multiple regression: correlation among predictor variables.

• Model Assumptions

  • This essentially means that the predictor variables $x$ can be treated as fixed values rather than random variables.
  • Because the predictor variables are treated as fixed values (see above), linearity is really only a restriction on the parameters.
  • The predictor variables themselves can be arbitrarily transformed, and in fact multiple copies of the same underlying predictor variable can be added, each one transformed differently.
  • Lack of multicollinearity in the predictors.
  • This can be triggered by having two or more perfectly correlated predictor variables (e.g. if the same predictor variable is mistakenly given twice, either without transforming one of the copies or by transforming one of the copies linearly).

• The chi-square distribution and finding areas

  • The cutoff 4.3 falls between the second and third columns in the 2 degrees of freedom row.
  • Looking in the row with 5 df, 5.1 falls below the smallest cutoff for this row (6.06).
  • Figure 6.9(d) shows a cutoff of 11.7 on a chi-square distribution with 7 degrees of freedom.
  • Figure 6.9(e) shows a cutoff of 10 on a chi-square distribution with 4 degrees of freedom.
  • Figure 6.9(f) shows a cutoff of 9.21 with a chi-square distribution with 3 df.

• Email data

  • The email data set was first presented in Chapter 1 with a relatively small number of variables.In fact, there are many more variables available that might be useful for classifying spam.Descriptions of these variables are presented in Table 8.13.The spam variable will be the outcome, and the other 10 variables will be the model predictors.While we have limited the predictors used in this section to be categorical variables (where many are represented as indicator variables), numerical predictors may also be used in logistic regression.
  • Recall from Chapter 7 that if outliers are present in predictor variables, the corresponding observations may be especially influential on the resulting model.

• Categorical predictors with two levels

  • Here we consider a categorical predictor with two levels (recall that a level is the same as a category).
  • For categorical predictors with just two levels, the linearity assumption will always be satisfied.
  • We'll elaborate further on this Ebay auction data in Chapter 8, where we examine the influence of many predictor variables simultaneously using multiple regression.
  • This is especially important since some of the predictors are associated.

• Making Inferences About the Slope

  • A fitted linear regression model can be used to identify the relationship between a single predictor variable, $x$, and the response variable, $y$, when all the other predictor variables in the model are "held fixed".
  • The meaning of the expression "held fixed" may depend on how the values of the predictor variables arise.
  • If the experimenter directly sets the values of the predictor variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been "held fixed" by the experimenter.
  • In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable.
  • However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variables when the predictors are correlated with each other and are not assigned following a study design.

• Introducing the t distribution

  • If we want the value in this row that identifies the cutoff for an upper tail of 10%, we can look in the column where one tail is 0.100.
  • This cutoff is 1.33.
  • If we had wanted the cutoff for the lower 10%, we would use -1.33.
  • The columns describe the cutoffs for specific tail areas.