Regression Statistics

Interpretation

Multiple R*

0.996

r= Coefficient of Simple Correlation = the positive square root of r-squared

R Square*

0.994 = 99%

r-squared = Coefficient of Simple Determination = percent of the variation in the y-variable that is explained by the x-variable

Adjusted R Square*

0.990

r-squared adjusted = version of r-squared that has been adjusted for the number of predictors in the model.  r-squared tends to over estimate the strength of the association, especially when there are more than one independent variables

Standard Error

1.244877

 

Observations

7

Number of paired data items – number of observations in the sample

*R-squared reduces to r-squared for simple linear regression when there is only one independent variable in the model.

 

r-squared = coefficient of simple determination

r-squared = 1 when all the observations fall directly on the fitted response surface

A large r-squared does not necessarily imply that the fitted model is a useful one.  One example of this occurs if the observations are taken at a narrow interval and the predictions are wanted outside the region of observations.  If the MSE is too large and high precision is required, then even though r-squared is large, the inferences may not be useful,

 

Adding more variables to a model can only increase r-squared and never reduce it.  This is because the SSE can never become larger with more independent variables and SSTO is always the same for a given set of responses.  Therefore we need an adjusted coefficient of multiple determination.

r-squared adjusted may actually become smaller when another independent variable is added to the model,  because the decrease in SSE may be more than offset by the loss of a degree of freedom in the denominator, n-p.

 

 

 

   

ANOVA

Interpretation

 

df

df = Degrees of Freedom

Regression

 

2

Number of independent variables in the model.  Abbreviated df(reg)

Residual

 

4

Number of observations – number of independent variables in the model – 1

Total

 

6

Number of observations – 1         Also, df(total) = df(reg) + df(residual)

 

 

ANOVA

 

 

Interpretation

 

df

SS

SS = Sum of Squares

Regression

2

970.6583

SS Regression = SS Reg

Residual

4

  6.198878

SS Residual = SS Error = SSE

Total

6

976.86

SS Total = SST = SS Regression + SS Residual

 

 

ANOVA

 

 

 

Interpretation

 

df

SS

MS

MS=Mean Square

Regression

1

905.53

905.53

MS Regression = SS Regression / df Regression = 905.53/1 =905.53

Residual

5

  71.33

   14.23

 

MS Residual = SS Residual / df Residual = 71.33/5 = 14.23

Also called MS Error = MSE

Total

6

976.86

 

MS Total = MST = MS Regression  +  MS Residual

 

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