Interpreting coefficients
Lucy D’Agostino McGowan
Interpreting coefficients
In general, in what context are we concerned with interpreting coefficients, prediction or inference?
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
Note: Always state what the units are for \(x\) and what the units are for \(y\) in the context of the problem
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
Note: Always state what the units are for \(x\) and what the units are for \(y\) in the context of the problem AND what the variables being held constant are
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)\) holding all other variables constant |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)\) holding all other variables constant |
| multiple linear regression where a quadratic and cubic term of \(x_i\) are also included |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)\) holding all other variables constant |
| multiple linear regression where a quadratic and cubic term of \(x_i\) are also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)+ \hat\beta_{x_i^3}(b^3-a^3)\) holding all other variables constant |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)\) holding all other variables constant |
| multiple linear regression where a quadratic and cubic term of \(x_i\) are also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)+ \hat\beta_{x_i^3}(b^3-a^3)\) holding all other variables constant |
| multiple linear regression where an interaction of \(x_i\) with a binary variable is included |
Interpreting coefficients
| model type | interpretation of \(\hat\beta\) |
|---|---|
| simple linear regression | A one unit change in \(x\) yields an expected change in \(y\) of \(\hat\beta_1\) |
| multiple linear regression where \(x_i\) is only included once in the model | A one unit change in \(x_i\) yields an expected change in \(y\) of \(\hat\beta_i\) holding all other variables constant |
| multiple linear regression where a quadratic of \(x_i\) is also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)\) holding all other variables constant |
| multiple linear regression where a quadratic and cubic term of \(x_i\) are also included | A change in \(x_i\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_{x_i}(b-a) + \hat\beta_{x_i^2}(b^2-a^2)+ \hat\beta_{x_i^3}(b^3-a^3)\) holding all other variables constant |
| multiple linear regression where an interaction of \(x_1\) with a variable \(x_{2}\) is included | A one unit change in \(x_1\) yields an expected change in \(y\) of \(\hat\beta_1\) when \(x_{2}=0\) holding all other variables constant |
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x\]
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x\]
How do you interpret \(\hat\beta_0\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2\]
How do you interpret \(\hat\beta_0\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2\]
How do you interpret \(\hat\beta_0\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2+\hat\beta_3x_2\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_1\times x_2\]
How do you interpret \(\hat\beta_0\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_1\times x_2\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_1\times x_2\]
How do you interpret \(\hat\beta_2\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_1\times x_2\]
How do you interpret \(\hat\beta_3\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_3 + \hat\beta_4x_1\times x_2\]
How do you interpret \(\hat\beta_0\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_3 + \hat\beta_4x_1\times x_2\]
How do you interpret \(\hat\beta_1\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_3 + \hat\beta_4x_1\times x_2\]
How do you interpret \(\hat\beta_2\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_3 + \hat\beta_4x_1\times x_2\]
How do you interpret \(\hat\beta_3\)?
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2 + \hat\beta_3x_3 + \hat\beta_4x_1\times x_2\]
How do you interpret \(\hat\beta_4\)?
