Lecture 16

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Lecture 16 Thurs, Oct. 30

• Inference for Regression (Sections 7.3-7.4)
• Hypothesis Tests and Confidence Intervals for
  Intercept and Slope
• Confidence Intervals for mean response
• Prediction Intervals
• Next time Robustness of least squares
  inferences, graphical tools for model assessment
  (8.1-8.3)

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Regression

• Goal of regression Estimate the mean response Y
  for subpopulations Xx,
• Example Y neuron activity index, Xyears
  playing stringed instrument
• Simple linear regression model
• Estimate and by least squares
  choose to minimize the sum of squared
  residuals (prediction errors)

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Ideal Model

• Assumptions of ideal simple linear regression
  model
• There is a normally distributed subpopulation of
  responses for each value of the explanatory
  variable
• The means of the subpopulations fall on a
  straight-line function of the explanatory
  variable.
• The subpopulation standard deviations are all
  equal (to
• )
• The selection of an observation from any of the
  subpopulations is independent of the selection of
  any other observation.
  

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The standard deviation

• is the standard deviation in each
  subpopulation.
• measures the accuracy of predictions from the
  regression.
• If the simple linear regression model holds, then
  approximately
• 68 of the observations will fall within of
  the regression line
• 95 of the observations will fall within of
  the regression line

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Estimating

• Residuals provide basis for an estimate of
• Degrees of freedom for simple linear regression
  n-2
• If the simple linear regression models holds,
  then approximately
• 68 of the observations will fall within of
  the least squares line
• 95 of the observations will fall within of
  the least squares line

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Inference for Simple Linear Regression

• Inference based on the ideal simple linear
  regression model holding.
• Inference based on taking repeated random samples
  ( ) from the same subpopulations
• ( ) as in the observed data.
• Types of inference
• Hypothesis tests for intercept and slope
• Confidence intervals for intercept and slope
• Confidence interval for mean of Y at XX0
• Prediction interval for future Y for which XX0

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Hypothesis tests for and

• Hypothesis test of vs.
  
• Based on t-test statistic,
• p-value has usual interpretation, probability
  under the null hypothesis that t would be at
  least as large as its observed value, small
  p-value is evidence against null hypothesis
• Hypothesis test for vs.
  is based on an analogous test statistic.
• Test statistics and p-values can be found on JMP
  output under parameter estimates, obtained by
  using fit line after fit Y by X.

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JMP output for example
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Confidence Intervals for and

• Confidence intervals provide a range of plausible
  values for and
• 95 Confidence Intervals
• Finding CIs in JMP Can find
  
• under parameter estimates after fitting line.
  Can find in Table A.2.
• For brain activity study, CIs

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Confidence Intervals for Mean of Y at XX0

• What is a plausible range of values for
  
• 95 CI for
  
• ,
  
• Note about formula
• Precision in estimating is not
  constant for all values of X. Precision
  decreases as X0 gets farther away from sample
  average of Xs
• JMP implementation Use Confid Curves fit command
  under red triangle next to Linear Fit after using
  Fit Y by X, fit line. Use the crosshair tool to
  find the exact values of the confidence interval
  endpoints for a given X0.

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Prediction Intervals

• What are likely values for a future value Y0 at
  some specified value of X (X0)?
• The best single prediction of a future response
  at X0 is the estimated mean response
• A prediction interval is an interval of likely
  values along with a measure of the likelihood
  that interval will contain response.
• 95 prediction interval for X0 If repeated
  samples are obtained from the
  subpopulations and a prediction
  interval is formed, the prediction interval will
  contain the value of Y0 for a future observation
  from the subpopulation X0 95 of the time.
  

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Prediction Intervals Cont.

• Prediction interval must account for two sources
  of uncertainty
• Uncertainty about the location of the
  subpopulation mean
• Uncertainty about where the future value will be
  in relation to its mean
• Prediction Error Random Sampling Error
  Estimation Error

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Prediction Interval Formula

• 95 prediction interval at X0
• Compare to 95 CI for mean at X0
• Prediction interval is wider due to random
  sampling error in future response
• As sample size n becomes large, margin of error
  of CI for mean goes to zero but margin of error
  of PI doesnt.
• JMP implementation Use Confid Curves Indiv
  command under red triangle next to Linear Fit
  after using Fit Y by X, fit line. Use the
  crosshair tool to find the exact values of the
  confidence interval endpoints for a given X0.

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Example

• A building maintenance company is planning to
  submit a bid on a contract to clean 40 corporate
  offices scattered throughout an office complex.
  The costs incurred by the maintenance company are
  proportional to the number of crews needed for
  this task. Currently the company has 11 crews.
  Will 11 crews be enough?
• Recent data are available for the number of rooms
  that were cleaned by varying number of crews.
  The data are in cleaning.jmp.
• Assuming a simple linear regression model holds,
  which is more relevant for answering the question
  of interest a confidence interval for the mean
  number of rooms cleaned by 11 crews or a
  prediction interval for the number of rooms
  cleaned on a particular day by 11 crews?

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Correlation

• Section 7.5.4
• Correlation is a measure of the degree of linear
  association between two variables X and Y. For
  each unit in population, both X and Y are
  measured.
• Population correlation
• Correlation is between 1 and 1. Correlation of
  0 indicates no linear association. Correlations
  near 1 indicates strong positive linear
  association correlations near 1 indicate strong
  negative linear association.

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Correlation and Regression

• Features of correlation
• Dimension-free. Units of X and Y dont matter.
• Symmetric in X and Y. There is no response and
  explanatory variable.
• Correlation only measures degree of linear
  association. It is possible for there to be an
  exact relationship between X and Y and yet sample
  correlation coefficient is zero.
• Correlation in JMP Click multivariate and put
  variables in Y, columns.
• Connection to regression
• Test of slope vs.
  is identical to test of vs.
  . Test of correlation coefficient
  only makes sense if the pairs (X,Y) are randomly
  sampled from population.

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Correlation in JMP