Hypothesis Testing

Hypothesis TestingDr. D’Agostino McGowan1 / 8

Dr. Lucy D'Agostino McGowan

AssumptionsIf you just want to estimate the parameters ββ, you don't have to make any distributional assumptions for ϵϵ. 
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Dr. Lucy D'Agostino McGowan

AssumptionsIf you just want to estimate the parameters ββ, you don't have to make any distributional assumptions for ϵϵ. 
This may sound different from what you learned in previous classes! The reason we have distributional assumptions is so that we can create confidence intervals and do hypothesis testing
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Dr. Lucy D'Agostino McGowan

AssumptionsSince we are doing least squares we assume the errors are independent and identically distributed with a mean of 0 and a variance of σ2σ2. 
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Assumptions

Since we are doing least squares we assume the errors are independent and identically distributed with a mean of 0 and a variance of $σ^{2}$ .
For hypothesis testing + confidence intervals, let's assume the distribution is normal:

$ϵ \sim N (0, σ^{2} I)$

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Assumptions

Since we are doing least squares we assume the errors are independent and identically distributed with a mean of 0 and a variance of $σ^{2}$ .
For hypothesis testing + confidence intervals, let's assume the distribution is normal:

$ϵ \sim N (0, σ^{2} I)$

Since $y = X β + ϵ$ , we know $y \sim N (X β, σ^{2} I)$

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Assumptions

Since we are doing least squares we assume the errors are independent and identically distributed with a mean of 0 and a variance of $σ^{2}$ .
For hypothesis testing + confidence intervals, let's assume the distribution is normal:

$ϵ \sim N (0, σ^{2} I)$

Since $y = X β + ϵ$ , we know $y \sim N (X β, σ^{2} I)$
From this it follows that

$\hat{β} = (X^{T} X)^{- 1} X^{T} y \sim N (β, (X^{T} X)^{- 1} σ^{2})$

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Dr. Lucy D'Agostino McGowan

Hypothesis testingOften we want to compare a small model to a larger one
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Dr. Lucy D'Agostino McGowan

Hypothesis testingOften we want to compare a small model to a larger one
Parsimony is great! We want to fit the smallest model that can best explain our outcome. But how do we determine if the small model is good enough?
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Dr. Lucy D'Agostino McGowan

Hypothesis testingOften we want to compare a small model to a larger one
Parsimony is great! We want to fit the smallest model that can best explain our outcome. But how do we determine if the small model is good enough?
We can compare the RSS for the two models!
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Dr. Lucy D'Agostino McGowan

Hypothesis testingOften we want to compare a small model to a larger one
Parsimony is great! We want to fit the smallest model that can best explain our outcome. But how do we determine if the small model is good enough?
We can compare the RSS for the two models!
If RSSsmall−RSSlargerRSSsmall−RSSlarger is small, then the fit of the small model is almost as good as the larger one!
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Hypothesis testing

The test statistic of interest is:

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}}$

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Hypothesis testing

The test statistic of interest is:

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}}$

How do we decide what value is "good enough"?

5 / 8

Hypothesis testing

The test statistic of interest is:

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}}$

How do we decide what value is "good enough"?

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}} > some constant$

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Hypothesis testing

The test statistic of interest is:

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}}$

How do we decide what value is "good enough"?

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}} > some constant$

What constant should we use?

5 / 8

Hypothesis testing

The test statistic of interest is:

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}}$

How do we decide what value is "good enough"?

$\frac{R S S_{s m a l l} - R S S_{l a r g e r}}{R S S_{l a r g e r}} > some constant$

What constant should we use?

Let's create an F-Statistic so we can compare to the F-distribtion!

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The F-Statistic

$F = \frac{R S S_{s m a l l} - R S S_{l a r g e r} / (d f_{s m a l l} - d f_{l a r g e r})}{R S S_{l a r g e r} / d f_{l a r g e r}}$

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The F-Statistic

$F = \frac{R S S_{s m a l l} - R S S_{l a r g e r} / (d f_{s m a l l} - d f_{l a r g e r})}{R S S_{l a r g e r} / d f_{l a r g e r}}$

If the small model has 3 variables and the larger model has 5, what are the degrees of freedom?

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The F-Statistic

$F = \frac{R S S_{s m a l l} - R S S_{l a r g e r} / (d f_{s m a l l} - d f_{l a r g e r})}{R S S_{l a r g e r} / d f_{l a r g e r}}$

If the small model has 3 variables and the larger model has 5, what are the degrees of freedom?
small: n - 4
larger: n - 6

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The F-Statistic

$F = \frac{R S S_{s m a l l} - R S S_{l a r g e r} / (d f_{s m a l l} - d f_{l a r g e r})}{R S S_{l a r g e r} / d f_{l a r g e r}} \sim F_{d f_{s m a l l} - d f_{l a r g e r}, d f_{l a r g e r}}$

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The F-Statistic

$F = \frac{R S S_{s m a l l} - R S S_{l a r g e r} / (d f_{s m a l l} - d f_{l a r g e r})}{R S S_{l a r g e r} / d f_{l a r g e r}} \sim F_{d f_{s m a l l} - d f_{l a r g e r}, d f_{l a r g e r}}$

We will reject the null hypothesis (that small = larger) if $F > F_{d f_{s m a l l} - d f_{l a r g e r}, d f_{l a r g e r}}^{α}$

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The F-Statistic

What if you want to compare your full model to an intercept only model?

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The F-Statistic

What if you want to compare your full model to an intercept only model?

$H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$

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The F-Statistic

What if you want to compare your full model to an intercept only model?

$H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$

What is RSS for the small model?

8 / 8

The F-Statistic

What if you want to compare your full model to an intercept only model?

$H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$

What is RSS for the small model?

8 / 8

The F-Statistic

What if you want to compare your full model to an intercept only model?

$H_{0} : β_{1} = β_{2} = \dots = β_{p} = 0$

What is RSS for the small model?

$F = \frac{(T S S - R S S) / p}{R S S / (n - (p + 1))}$

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Hypothesis Testing

Dr. D’Agostino McGowan

Assumptions

Assumptions

Assumptions

Assumptions

Assumptions

Assumptions

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

Hypothesis testing

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

The F-Statistic

Assumptions

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