Interpreting coefficients
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 \(\hat\beta_0\)?
“When \(x=0\) the average value for y is \(\hat\beta_0\)”
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x\]
How do you interpret \(\hat\beta_1\)?
“For every one unit increase in x the expected change in y is \(\hat\beta_1\)”
Examples
With an indicator variable
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1(x == 1)\] ::: question How do you interpret \(\hat\beta_0\)? :::
“When \(x=0\) the expected \(y\) is equal to \(\hat\beta_0\)”
Examples
With an indicator variable
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1(x == 1)\] ::: question How do you interpret \(\hat\beta_1\)? :::
“Being in group \(x=1\) compared to \(x=0\) yields an expected change in y of \(\hat\beta_1\)”
Examples
With an indicator variable, multiple categories, referent is \(x = C\)
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1(x == A) + \hat{\beta}_2(x == B)\] ::: question How do you interpret \(\hat\beta_0\)? :::
“When \(x=C\) the average value of y is \(\hat\beta_0\)”
Examples
With an indicator variable, multiple categories, referent is \(x = C\)
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1(x == A) + \hat{\beta}_2(x == B)\] ::: question How do you interpret \(\hat\beta_1\)? :::
“Being in group \(x = A\) compared to the referent category (\(x = C\)) yields an expected change in y of \(\hat\beta_1\)”
Examples
With an indicator variable, multiple categories, referent is \(x = C\)
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1(x == A) + \hat{\beta}_2(x == B)\]
How do you interpret \(\hat\beta_2\)?
“Being in group \(x = B\) compared to the referent category (\(x = C\)) yields an expected change in y of \(\hat\beta_2\)”
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2\]
How do you interpret \(\hat\beta_0\)?
“When \(x_1=0\) and \(x_2 = 0\), the average value for y is \(\hat\beta_0\)”
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_2\]
How do you interpret \(\hat\beta_1\)?
“For every one unit increase in x the expected change in y is \(\hat\beta_1\) holding \(x_2\) constant”
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2\]
How do you interpret \(\hat\beta_0\)?
“When \(x_1=0\), the average value for \(y\) is \(\hat\beta_0\)”
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2\]
How do you interpret \(\hat\beta_1\)?
We cannot interpret \(\hat\beta_1\) independent of \(\hat\beta_2\). \(\hat\beta_1\) is the linear term for \(x_1\) and \(\hat\beta_2\) is the quadratic term for \(x_1\), a change in \(x_1\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_1(b-a)+\hat\beta_2(b^2-a^2)\).
Examples
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat\beta_2x_1^2+\hat\beta_3x_2\]
How do you interpret \(\hat\beta_1\)?
We cannot interpret \(\hat\beta_1\) independent of \(\hat\beta_2\). \(\hat\beta_1\) is the linear term for \(x_1\) and \(\hat\beta_2\) is the quadratic term for \(x_1\), a change in \(x_1\) from \(a\) to \(b\) yields an expected change in \(y\) of \(\hat\beta_1(b-a)+\hat\beta_2(b^2-a^2)\) holding \(x_2\) constant.
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 \(\hat\beta_0\)?
“When \(x_1=0\) and \(x_2=0\), the average value for \(y\) is \(\hat\beta_0\)”
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 \(\hat\beta_1\)?
“When \(x_2=0\), for every one unit increase in \(x_1\) the expected change in y is \(\hat\beta_1\)”
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 \(\hat\beta_2\)?
“When \(x_1=0\), for every one unit increase in \(x_2\) the expected change in y is \(\hat\beta_2\)”
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 \(\hat\beta_3\)?
“For every one unit increase in \(x_1\) the expected change in slope for \(x_2\) is \(\hat\beta_3\)”
“For every one unit increase in \(x_2\) the expected change in slope for \(x_1\) is \(\hat\beta_3\)”
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 \(\hat\beta_0\)?
“When \(x_1=0\), \(x_2=0\), and \(x_3=0\), the average value for \(y\) is \(\hat\beta_0\)”
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 \(\hat\beta_1\)?
“When \(x_2=0\), for every one unit increase in \(x_1\) the expected change in y is \(\hat\beta_1\) holding \(x_3\) constant”
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 \(\hat\beta_2\)?
“When \(x_1=0\), for every one unit increase in \(x_2\) the expected change in y is \(\hat\beta_2\) holding \(x_3\) constant”
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 \(\hat\beta_3\)?
“For every one unit increase in \(x_3\) the expected change in y is \(\hat\beta_3\), holding \(x_1\) and \(x_2\) constant”
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 \(\hat\beta_4\)?
“For every one unit increase in \(x_1\) the expected change in slope for \(x_2\) is \(\hat\beta_4\) holding \(x_3\) constant”
“For every one unit increase in \(x_2\) the expected change in slope for \(x_1\) is \(\hat\beta_4\) holding \(x_3\) constant”
