Regression lecture 2

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Regression lecture 2

• 1. Review deterministic and random components
• 2. The coefficient of determination
• 3. Using the regression line
• 4. Estimation of the mean value of Y for some X
• 5. Prediction of an individual value of Y for
  some X
• 6. Estimation and prediction contrasted
• 7. Estimation and prediction formulas
• 8. Examples

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1. Deterministic random components

• Our basic question is whether there is a
  relationship between two variables, X and Y.
• To answer this question, we compare the
  deterministic part of the relationship to the
  random part.
• the deterministic part is the part that would
  look the same if we sampled and measured again
  its there for a reason (if its there at all).

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1. Deterministic random components

• Deterministic part the least squares line
  (which determines a Y for each value of X).
• Random part deviations of observed Y scores
  from least squares line
• (Note similarity here to t, F, and Z tests, where
  we compare the numerator (treatment error) to
  the denominator of (error).)

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2. Coefficient of determination

• Once we have regression line, we can assess its
  usefulness as a numerical model of the X-Y
  relationship.
• We can do this by testing a hypothesis about the
  slope ß1 or the correlation ?, as last week.
• We can also square the correlation coefficient,
  to get the coefficient of determination, r2.

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The sum of the line (---) lengths gives the
total error when we compute the Yi using the
mean, Y
Y
X
SSYY S(Yi Y)2
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Regression line
Y
The sum of the line (---) lengths gives the total
error when we compute the Yi using the regression
line.
X

SSE S(Yi Yi)2
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2. The coefficient of determination

• If knowing X reduces our uncertainty about Y,
  then SSE ltlt SSYY. In that case, r2 the
  coefficient of determination tells us something
  useful
• SSYY SSE
• SSYY
• r2 explained sample variability in Y
• total sample variability in Y

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3. Using the regression line

• So far, weve learned how to decide whether our
  regression line is useful.
• Suppose the test of hypothesis tells us the line
  is useful. What can we do with it?
• Well consider two alternative uses estimation
  and prediction.

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3. Using the regression line

• Estimation
• gives the average value of Y (Y) for all cases
  that have a given value of X
• Prediction
• gives an individual Y score for one case that
  has a given value of X

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4. Estimation of the mean value of Y for some X

• We can estimate the mean value of Y for a
  specific value of X.
• e.g., we can estimate Y for ALL people whose
  blood contains a 4 concentration of some drug
• here, Y would be some variable of interest such
  as (for example) reaction time (RT) to perform
  some task
• we could estimate mean RT for all people who
  have the 4 drug concentration in their blood

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5. Prediction of an individual value of Y for
some X

• We can predict an individual value of Y for a
  given value of X.
• e.g., we could predict RT for a specific person
  whose blood contains a 4 concentration of the
  drug

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6. Estimation and prediction contrasted

• Recall from last week the two sources of error
  when using X to calculate an expected Y
• 1. In the population, Y is not uniquely
  determined by X. As a result, for each value of
  X, there is a distribution of possible Y values.
• if we knew the line Y ß0 ß1X e, we would
  still have this source of error

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6. Estimation and prediction contrasted

• Two sources of error when using X to calculate an
  expected value of Y
• 2. The line we do have, Y ß0 ß1X, is not
  precisely correct
• it does not capture the relationship between X
  and Y very precisely, because it is based on
  sample data.

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6. Estimation and prediction contrasted

• Estimation
• only the second source of error is at work
• things other than X that influence Y in the
  population are random effects, so on average
  across all cases they cancel out
• Predicting
• both sources of error are at work

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7. Estimation and prediction formulas

• Estimation interval
• Y (ta/2)(s) 1 (XP X)2
• n SSXX
• ta/2 is based on d.f. n 2

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7. Estimation and prediction formulas

• Prediction interval
• Y (ta/2)(s) 1 1 (XP X)2
• n SSXX
• ta/2 is based on d.f. n 2

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Examples Emotional intelligence

• First, we find X and Y
• X SX 74 10.571
• n 7
• Y SY 82 11.714
• n 7

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Examples Emotional intelligence

• From last week
• SSXY 109.143
• SSXX 139.71
• Thus, ß1 109.143 .781
• 139.71

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Examples Emotional intelligence

• ß0 Y ß1X
• 11.714 .781(10.571)
• 3.46
• SSE SSYY ß1(SSXY)
• 115.429 .781 (109.143)
• 30.188

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Examples Emotional intelligence

• S SSE
• n 2
• 30.188
• 5
• 2.457

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Examples Emotional intelligence

• The question says Use the data to form a 95
  prediction interval for the Openness score of
  someone with an EI score of 13.
• Y ß0 ß1(X) 3.46 .781 (13) 13.613
• tcrit t(5, a/2 .025) 2.571.

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Examples Emotional intelligence

• Interval is
• 13.613 (2.571) (2.457) 1 1 (13
  10.571)2
• 7 139.71
• 13.613 6.877

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Examples Laughing

• First, we find X and Y
• X SX 4.2 .60
• n 7
• Y SY 32 4.5714
• n 7

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Examples Laughing

• From last week
• SSXY 2.15
• SSXX .34
• Thus, ß1 2.15 6.3235
• .34

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Examples Laughing

• ß0 Y ß1X
• 4.5714 6.3235(.60)
• .7773
• SSE SSYY ß1(SSXY)
• 15.2143 6.3235 (2.15)
• 1.6188

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Examples Laughing

• S SSE
• n 2
• 1.6188
• 5
• .569

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Examples Laughing

• The question says Regardless of your answer to
  part (a), form the 95 confidence interval for
  the predicted y value for a delay of .5 seconds
  (i.e., for all instances of .5).
• Y ß0 ß1(X) .7773 6.3235 (.5) 3.939
• tcrit t(5, a/2 .025) 2.571.

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Examples Laughing

• Interval is
• 3.939 (2.571) (.569) 1 (.5 .6)2
• 7 .34
• 3.939 (2.571) 9.569) (.4151)
• 3.939 .607

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