SMHS LinearModeling MLR VizModelCoeff - Revision history

Imoubara: /* Protocols for Rectifying Residual Plots Indicating Nonlinearity */

2016-05-23T13:57:02Z

‎Protocols for Rectifying Residual Plots Indicating Nonlinearity

2016-05-23T13:26:55Z

‎Independence

2016-05-20T15:45:44Z

‎Lack of collinearity

2016-05-20T15:45:35Z

‎Protocols for rectifying residual plots indicating nonlinearity

2016-05-20T15:44:33Z

‎Outliers (influential data points)

2016-05-20T15:44:06Z

‎Outliers (influential data points)

2016-05-20T14:27:47Z

‎Outliers (influential data points)

2016-05-20T14:27:37Z

‎Normality of residuals

2016-05-20T14:27:27Z

‎Heteroskedasticity (lack of homoscedasticity)

2016-05-20T14:27:19Z

‎Lack of collinearity

@@ Line 83: / Line 83: @@
 Model may be excluding important fixed effects that interact with fixed effects already accounted for in the model. If new fixed effects are added, the pattern in the residual plot may disappear.
-<li>Variable transformation – Perform a nonlinear transformation of the response, e.g., log, or reciprocal transform. See this SOCR Activity<sup>5</sup>.
+*Variable transformation – Perform a nonlinear transformation of the response, e.g., log, or reciprocal transform. See this SOCR Activity<sup>5</sup>.
-<li>Perform a nonlinear transformation to explanatory variables. For instance, if Age is related to Weight in a U-shaped way (e.g., quadratic relation), then the model could include Age and Age<sup>2</sup> as predictors.
+*Perform a nonlinear transformation to explanatory variables. For instance, if Age is related to Weight in a U-shaped way (e.g., quadratic relation), then the model could include Age and Age<sup>2</sup> as predictors.
-<li>If residual plots include stripes, then there may be categorical variables playing role, which may necessitate alternative class of models, such as logistic or multinomial models.
+*If residual plots include stripes, then there may be categorical variables playing role, which may necessitate alternative class of models, such as logistic or multinomial models.
 ===Lack of Collinearity===

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 Independence is mostly a question of the experimental design which is tightly intertwined with the subsequent statistical analyses. When the study design demands collection of more data per subject, repeated measure designs, the data will have (time) dependencies and the appropriate statistical methodologies for interrogating such data involve mixed linear models.
 <sup>5</sup> http://wiki.socr.umich.edu/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs<BR>
-<sup>6</sup>http://socr.umich.edu/html/dist/
+<sup>6</sup> http://socr.umich.edu/html/dist/
 ===See next===

@@ Line 87: / Line 87: @@
 <li>If residual plots include stripes, then there may be categorical variables playing role, which may necessitate alternative class of models, such as logistic or multinomial models.
-===Lack of collinearity===
+===Lack of Collinearity===
 A pair of (linearly) correlated predictors are also called collinear. Suppose Age and Height are correlated (which is certainly to be expected!), then including both as predictors of Weight presents a collinearity problem. In the presence of significant variable collinearity, the interpretation of the model becomes challenging. Depending on which correlated predictors are included in the model, the fixed effects may (incorrectly) become significant or insignificant. Significant findings for these correlated or collinear fixed effects is difficult to interpret as there may be trading-off between the "explanatory power" of the pair of covariates. Multiple predictors that are very similar (linearly correlated) inhibit our ability to decide what plays a big role and what has only marginal impact on the response.

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 <center>[[Image:SMHS_LinearModeling_Fig20.png|500px]]</center>
-===Protocols for rectifying residual plots indicating nonlinearity===
+===Protocols for Rectifying Residual Plots Indicating Nonlinearity===
 Model may be excluding important fixed effects that interact with fixed effects already accounted for in the model. If new fixed effects are added, the pattern in the residual plot may disappear.

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 Interpretation of large or a small DFbeta values is similarly "open" as the interpretation of p-values (case-specific). However, if the DFbeta value changes, the sign of the coefficient slope, then this data point is an outlier candidate (influential point) that needs special attention. Excluding that 1 point would change the interpretation of the model results.
-For small to medium size (n) data, |DFbeta| > 1 is generally suspicious (outlier).  For larger datasets, a rule of thumb criterion is |''DFbeta''| > <sup>2</sup>&frasl;<sub>&radic;<span style="text-decoration: overline">n</span></sub>
+For small to medium size (''n'') data, |DFbeta| > 1 is generally suspicious (outlier).  For larger datasets, a rule of thumb criterion is |''DFbeta''| > <sup>2</sup>&frasl;<sub>&radic;<span style="text-decoration: overline">n</span></sub>
   par(mfrow=c(1,1))

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 The histogram (left) is relatively bell-shaped and the Q-Q normal probability plot (right) indicates the residuals are mostly on a straight line (suggesting errors (observed-predicted) are similar to a normal distribution). Thus, there is no strong evidence suggesting possible violation of the normality assumption.
-====Outliers (influential data points)====
+===Outliers (influential data points)===
 Data should not include extreme influential points, otherwise the linear model may be heavily biased. Outliers (influential data points) can drastically change the interpretation of model results and inference, similarly to variable collinearity.

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 In this example, larger fitted values correspond with larger residuals (the opposite may be true as well). Transforming the data often helps resolve such Heteroskedasticity.
-====Normality of residuals====
+===Normality of residuals===
 The normality of residuals assumption (parametric assumptions) is also important. Linear models may be robust and gracefully handle certain some violations of the normality assumption.

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 Collinearity may be avoided early during the study-design/data-collection phase of the study to collect fewer fixed effects known to not be linearly correlated. Alternatively, we can selectively include/exclude predictors (e.g., only include in model the most meaningful independent variable and drop the others). Dimensionality reduction methods (e.g., Principal/Independent Component Analyses) also identify (fuse) linearly correlated variables into factors (representing linear combinations transforming several correlated variables into one component variable, which can be used as new fixed effect).
-====Heteroskedasticity (lack of homoscedasticity)====
+===Heteroskedasticity (lack of homoscedasticity)===
 If the variance of the data is (approximately) stable across the range of the predicted values, then the process is homoscedastic (equal variance assumption). Otherwise, when the homoscedasticity criterion is violated, the process is called heteroskedastic (unequal variances).