LESSON 4.4. MULTIPLE LINEAR REGRESSION. Residual Analysis

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

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Type of residuals

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

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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.

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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.

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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.

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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.

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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.

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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.

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Graphics of residuals
residuals
It implies the existence of a hidden variable.
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Graphics of residuals
Dot-plot of a group of residuals.
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Graphics of residuals-predictions
residuals
predictions
There is no problem detected.  
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Graphics of residuals-predictions
residuals
predictions
The linear fitness is not adequate.
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Graphics of residuals-predictions
residuals
predictions
Linear fitness wrongly calculated.
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Graphics of residuals-predictions
residuals
predictions
There is heteroscedasticity.
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Graphics of residuals-predictions
residuals
predictions
Non-linear fitness and heteroscedasticity.
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Graphics of residuals-predictions
residuals
predictions
There are some outliers.