LESSON 4.4. MULTIPLE LINEAR REGRESSION. Residual Analysis Design and Data Analysis in Psychology II Susana Sanduvete Chaves Salvador Chacón Moscoso

Type of residuals Residuals (ordinary): difference between the observation (Y) and prediction( ). The in residue ei is a random variable has the following properties : Under the assumption of normality is obtained:

Type of residuals Standardized residuals: errors after being established (zero mean and variance close to 1). Helps to distinguish huge residuals.

Type of residuals Outlier: one that has a large residue. Subjective criteria. The most common is to consider an outlier when its standardized residual is bigger than 2. The larger the standardized residual, more unusual is the observation.

Type of residuals Outliers are important because their inclusion or not in the sample can differ greatly estimated regression line. It is necessary to study direct scores with high standardized residuals. There are many causes that prompt the existence of outliers. Some of them are: The observed point is an error (in measurement, in the transcription of data, etc.), but the fitted model is adequate. The observed point is correct but the model fit is not, due to possible different reasons: Because the relationship between the two variables is linear in a certain range but it is not linear to the point where it is observed. There is a strong heteroscedasticity with some observations that are separated from the tag. There is a classification variable that has not been taken into account.

Type of residuals Studentized Residual: It is calculated the same way as standardized, but calculating the residual variance (sR) from the whole sample, except the residue of the observation under study. Thus, dependence between numerator and denominator disappears.

Type of residuals If n is high, the standardized and studentized residuals acquire close values. Under the normality hypothesis, it is verified that ti follows a t distribution with n- 3 degrees of freedom.

Type of residuals Eliminated residuals: Difference between the value observed in the answer and the prediction, when the whole sample is used, except the measurement that is being studied. If the measurement has a huge influence in the calculation of the regression line, the ordinary and eliminated residuals are different; in other cases, both values will be similar.

Graphics of residuals The Box-Plot and the histogram of standardized residuals provide information about their distribution. If the sample size is low, instead the histogram of residuals the dot-plot or the stem and leaf plot are used; their interpretations are the same.

Graphics of residuals It implies the existence of a hidden variable.

Graphics of residuals Dot-plot of a group of residuals.

Graphics of residuals-predictions There is no problem detected.  

Graphics of residuals-predictions The linear fitness is not adequate.

Graphics of residuals-predictions Linear fitness wrongly calculated.

Graphics of residuals-predictions There is heteroscedasticity.

Graphics of residuals-predictions Non-linear fitness and heteroscedasticity.

Graphics of residuals-predictions There are some outliers.