ANCOVA model

Examples of ANCOVA model in the following topics:

• Models with Both Quantitative and Qualitative Variables

  • A regression model that contains a mixture of quantitative and qualitative variables is called an Analysis of Covariance (ANCOVA) model.
  • A regression model that contains a mixture of both quantitative and qualitative variables is called an Analysis of Covariance (ANCOVA) model.
  • ANCOVA models are extensions of ANOVA models.
  • Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression.
  • To see if the CV significantly interacts with the IV, run an ANCOVA model including both the IV and the CVxIV interaction term.

• Two Regression Lines

  • Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression.
  • Therefore, when performing ANCOVA, we are adjusting the DV means to what they would be if all groups were equal on the CV.
  • Another use of ANCOVA is to adjust for preexisting differences in nonequivalent (intact) groups.
  • There are five assumptions that underlie the use of ANCOVA and affect interpretation of the results:
  • This pie chart shows the partitioning of variance within ANCOVA analysis.

• Comparing Nested Models

  • Multilevel models, or nested models, are statistical models of parameters that vary at more than one level.
  • These models can be seen as generalizations of linear models (in particular, linear regression); although, they can also extend to non-linear models.
  • Furthermore, multilevel models can be used as an alternative to analysis of covariance (ANCOVA), where scores on the dependent variable are adjusted for covariates (i.e., individual differences) before testing treatment differences.
  • Multilevel models are able to analyze these experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA.
  • Random slopes model.

• Introduction to Model selection

  • The best model is not always the most complicated.
  • In this section we discuss model selection strategies, which will help us eliminate from the model variables that are less important.
  • In this section, and in practice, the model that includes all available explanatory variables is often referred to as the full model.
  • Our goal is to assess whether the full model is the best model.
  • If it isn't, we want to identify a smaller model that is preferable.

• Two model selection strategies

  • If one of these smaller models has a higher adjusted R2 than our current model, we pick the smaller model with the largest adjusted R2.
  • That is, we fit the model including just the cond new predictor, then the model including just the stock photo variable, then a model with just duration, and a model with just wheels.
  • Each of the four models (yes, we fit four models!
  • We fit three new models:
  • If one of these models has a larger than the model with no variables, we use this new model.

• Model selection exercises

  • Determine if any other variable(s) should be removed from the model.
  • Determine if any other variable(s) should be removed from the model.
  • Based on this table, which variable should be added to the model first?
  • Based on this table, which variable should be added to the model first?
  • We should consider removing this variable from the model.

• A single-variable model for the Mario Kart data

  • Let's fit a linear regression model with the game's condition as a predictor of auction price.
  • The model may be written as:
  • Interpret the coefficient for the game's condition in the model.
  • So 10.90 means that the model predicts an extra $10.90 for those games that are new versus those that are used.
  • Summary of a linear model for predicting auction price based on game condition.

• Identifying variables in the model that may not be helpful

  • H0 : βi = 0 when the other explanatory variables are included in the model.
  • We might consider removing the stock photo variable from the model.
  • The adjusted R2 may be used as an alternative to p-values for model selection, where a higher adjusted R2 represents a better model fit.
  • For instance, we could compare two models using their adjusted R2 , and the model with the higher adjusted R2 would be preferred.
  • The fit for the full regression model, including the adjusted R2 .

• Stepwise Regression

  • The frequent practice of fitting the final selected model, followed by reporting estimates and confidence intervals without adjusting them to take the model building process into account, has led to calls to stop using stepwise model building altogether -- or to at least make sure model uncertainty is correctly reflected.
  • Forward selection involves starting with no variables in the model, testing the addition of each variable using a chosen model comparison criterion, adding the variable (if any) that improves the model the most, and repeating this process until none improves the model.
  • A way to test for errors in models created by stepwise regression is to not rely on the model's $F$-statistic, significance, or multiple-r, but instead assess the model against a set of data that was not used to create the model.
  • This is often done by building a model based on a sample of the dataset available (e.g., 70%) and use the remaining 30% of the dataset to assess the accuracy of the model.
  • Models that are created may be too-small than the real models in the data.

• Checking model assumptions using graphs

  • "All models are wrong, but some are useful" -George E.P.
  • The truth is that no model is perfect.
  • However, even imperfect models can be useful.
  • Reporting a flawed model can be reasonable so long as we are clear and report the model's shortcomings.
  • This video covers key ideas for evaluating a multiple regression model in the context of the model fit in Sections 8.1 and 8.2.