Lecture 14 Psychology 790
Chapter 2: Inferences in Regression andCorrelation Analysis
Lecture 14October 26, 2006Psychology 790
Overview Todays Lecture Schedule
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Todays Lecture
Inferences (aka Hypothesis Tests and confidence intervals)for:
1
0
Y
ANOVA tables for inferences.
Lecture 14 Psychology 790
Our New Schedule
Date Topic Chapter10/26 Inferences in regression and correlation analysis K 2
10/31 Diagnostics and remedial measures K 3
11/2 No Class - No Lab11/4 Simultaneous inferences and other topics K 4
11/7 Matrix algebra K 5
11/9 Multiple regression I K 6
11/14 Multiple regression II K 7
11/16 Regression models for quantitative and qualitative predictors K 8
11/21 No Class11/23 No Class - No Lab11/28 Building the regression model I K 9
11/30 Building the regression model II K 10
12/5 Building the regression model III (Final Handed Out) K 11
12/7 Open
Lecture 14 Psychology 790
Introductory Example
Overview
IntroductoryExample Snow geese Regression
Analysis
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Snow Geese
From Weisberg (1985, p. 102):
Aerial survey methods are regularly used to estimated thenumber of snow geese in their summer range areas west ofHudson Bay in Canada. To obtain estimates, small aircraft flyover the range and, when a flock of geese is spotted, anexperienced person estimates the number of geese in theflock. To investigate the reliability of this method of counting,an experiment was conducted in which an airplane carryingtwo observers flew over 45 flocks, and each observer made anindependent estimate of the number of birds in each flock.Also, a photograph of the flock was taken so that an exactcount of the number of birds in the flock could be made (databy Cook and Jacobsen, 1978).
Lecture 14 Psychology 790
Snow Geese
Lecture 14 Psychology 790
Hudson Bay
Overview
IntroductoryExample Snow geese Regression
Analysis
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Regression Analysis
Using the first observer in the plane, we consider therelationship between this persons count and that from thephotograph:
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observer 1 count
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100
200
300
400
photo
count
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Lecture 14 Psychology 790
Building on the Simple LinearRegression Model
Overview
IntroductoryExample
Building on ourModel
ErrorAssumptions
Our FirstAssumption
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
The Model
As seen previously...
The linear regression model (for observation i = 1, . . . , N ):
Yi = 0 + 1Xi + i
0 is the mean of the population when X is zero...the Yintercept.
1 is the slope of the line, the amount of increase in Ybrought about by a unit increase (X = X + 1) in X .
i is the random error, specific to each observation.
Overview
IntroductoryExample
Building on ourModel
ErrorAssumptions
Our FirstAssumption
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Lets talk about
Before we go any further, we need to talk about this crazy thing.
What do we know?
We know it is a random variable.
We know it is random error.
We know it has a mean of 0.
We know it has a variance of 2.
We can compute the regression line no matter thedistribution of the error term ().
However, we must have a distribution in order to dohypothesis testing.
Overview
IntroductoryExample
Building on ourModel
ErrorAssumptions
Our FirstAssumption
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Our First Assumption
With any statistical test, you must have a distribution.
Recall how we find critical values, p-values, etc...
Because we only know the mean and the variance of our ,we must make an assumption about the distribution of .
For most of this class, we will assume the following about theerror terms:
i N(0, 2)
Overview
IntroductoryExample
Building on ourModel
ErrorAssumptions
Our FirstAssumption
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Our First Assumption
We are going to use this assumption for the rest of the year.
It is a common theme to any regression model or generallinear model.
So, why assume that the error terms are normallydistributed?
With our new assumption of normal error terms, we can nowdo the following:
We now know the conditional distribution ofYi N(0 + 1Xi,
2), which can allow us to computeCIs.
We can now do great things with our s, namely computeconfidence intervals and perform hypothesis tests.
Overview
IntroductoryExample
Building on ourModel
ErrorAssumptions
Our FirstAssumption
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Lets Put Our Assumption to Good Use
So whats next?
Tests for the parameters, both 0 and 1.
Prediction intervals for the predicted value E(Yi).
Prediction interval for a point estimate (new observation).
Confidence bands for our regression line.
We are going to take a second approach to testing theparameters by using an ANOVA approach.
Talk about descriptive measures of the linear model, namelyr and r2.
Lecture 14 Psychology 790
Parameter Testing
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Parameter Testing
So let us begin with the parameter of most interest to us inour regression equation, the slope, 1.
We are going to use our regular old hypothesis that welearned, but we are going to change everything to relate toour 1.
Ok, last class we found out how to obtain a parameterparameter estimate, (now called b1 - drop the hat - the bookonly uses it for Y).
We need to know the distribution of that parameter in orderto perform a test.
This is where our assumption comes into play.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Parameter Testing - Test Statistic
Given our assumption of normal error terms, it follows that:
b1 N(1, 2(1)) where 2(1) =
2
(x x)2
So we have the distribution of 1, we need a test statistic forour hypothesis test.
T =b1 1s(b1)
t(n2) where s(b1) =MSE
(x x)2
Where do the degrees of freedom come from?
In regression, we lose one degree of freedom for eachparameter we estimate.
In this case we have two, 0 and 1, hence n 2.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Parameter Testing - 5 Step Process
1. First we need a null and alternative, we usually just want totest if there is any slope at all to the line:
H0 : 1 = 0
HA : 1 6= 0
2. Type I error rate () - set prior to the experiment.
3. Test statistic, computed using SAS, find it on the output.
4. Then decide to reject/fail to reject.
5. Interpret your results.
Significant 1 means there is significant linear relationshipbetween variables X and Y .
CODE:libname geese C:\data files\;
proc glm data=geese.geese;model photo=observer1;run;
OUTPUT:
18-1
Lecture 14 Psychology 790
Geese ExampleThe GLM Procedure
Dependent Variable: photo photo count
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 1 254769.4600 254769.4600 129.20 F
observer1 1 254769.4600 254769.4600 129.20 |t|
Intercept 26.64957256 8.61448190 3.09 0.0035
observer1 0.88255688 0.07764394 11.37
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Hypothesis Test
Estimated model: Y = 26.65 + 0.88X
H0 : 1 = 0
HA : 1 6= 0
= .05
test statisticT = 11.37(43df), p < 0.0001
Reject Null
We can conclude that there is a significant linear associationbetween the observer count and the true number of geese.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Confidence Interval
Again, the concept of a confidence interval for 1 is the sameas the other parameters we learned about, it is a bandaround which we are confident our true parameter falls into.
Our formula in this case is:
1 t(n 2, /2) (s(b1))
Of course...why compute yourself when you can use SAS?
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
CI in SAS
SAS will do the CI for you!
Code:
proc glm data=geese.geese;
model photo=observer1 / clparm;
run;
Parameter Output Portion:
Standard
Parameter Estimate Error t Value Pr > |t| 95% Confidence Limits
Intercept 26.64957256 8.61448190 3.09 0.0035 9.27681411 44.02233100
observer1 0.88255688 0.07764394 11.37
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
What about poor old 0?
Unfortunately for 0 it really isnt that useful for us in term ofhypothesis testing.
Why? Well, what would we be testing?
We would be testing if the mean of Y at X = 0 is equal tosome value.
First of all, this doesnt usually make sense to do.
Second, this test is only valid if the range of X includes 0.
Subnote: The Regression Equation is only valid over thespan of your range of value contained in your sample, notoutside of those values
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Lets Test 0 Anyway
Lets take pity on 0 and show how it can be done.
0 N(0, 2(0)) where 2(0) = 2
(
1
n+
x2
(x x)2
)
T =b0 0s(b0)
t(n2) where s(b0) = MSE(
1
n+
x2
(x x)2
)
The CI follows suit:
0 t(n 2, /2) (s(b0))
Or use SAS!
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
Hypothesis Test - From Example Again
H0 : 0 = 0
Ha : 0 = 0
= .05
test statistic
t(23) = 3.09(43df), p = 0.0035
Reject Null
We can conclude that the mean count of geese actuallypresent when no geese were observed is not zero.
Confidence Interval: (9.277, 44.022).
We are 95% confident that the true mean of geese presentwhen none were observed is between 9.277 and 44.022.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing Test for 1 Parameter Testing
- 5 Step Process 1 Example 1 CI About 0 Testing 0 Inference Notes
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping UpLecture 14 Psychology 790
A Note when making Inferences on s
What happens when you have deviations from normality?
The hypothesis tests are fairly robust to minor deviationsfrom normality.
We will talk later about how to test the normalityassumption of [like, next class]...
Power - The formula for power is in the book and it is verymessy. Anyone interested can ask me about it later.
Lecture 14 Psychology 790
Interval Estimation of Predictions
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions Example Mean
Interval Intervals for Y Example Mean
Interval Plots
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
The distribution of E(Y )
Before we begin any talk about CI, we always need to startwith a distribution of the parameter, that is what determinesour CI.
So our new parameter is E(Yh)
Yh N(0 + 1 Xh, 2
(
1
n+
(Xh x)2
(x x)2
)
)
distributed t(n 2), so CI is:
Yh t(n 2, /2) s(Yh)
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions Example Mean
Interval Intervals for Y Example Mean
Interval Plots
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
From our example...
Want to build a 95% CI for predicted value for an observationof 30 geese.
First, we need to find Y for X = 30.
Y30 = 26.65 + 0.88(30) = 48.71
Next, we find our standard deviation:
s(Yh) = . . .
Then we get tired and use:proc glm data=geese.geese;
model photo=observer1 / clm;
run;
Lecture 14 Psychology 790
From our example...The SAS System 00:05 Wednesday, October 25, 2006 13
The GLM Procedure
95% Confidence Limits for
Observation Observed Predicted Residual Mean Predicted Value
1 56.0000000 70.7774166 -14.7774166 57.0286892 84.5261440
2 38.0000000 48.7134946 -10.7134946 33.5445777 63.8824115
3 25.0000000 53.1262790 -28.1262790 38.3130930 67.9394650
4 48.0000000 57.5390634 -9.5390634 43.0480025 72.0301243
5 38.0000000 48.7134946 -10.7134946 33.5445777 63.8824115
6 22.0000000 44.3007102 -22.3007102 28.7447619 59.8566584
7 22.0000000 37.2402551 -15.2402551 21.0056060 53.4749042
8 42.0000000 56.6565065 -14.6565065 42.1038195 71.2091935
9 34.0000000 44.3007102 -10.3007102 28.7447619 59.8566584
10 14.0000000 35.4751414 -21.4751414 19.0602616 51.8900211
11 30.0000000 48.7134946 -18.7134946 33.5445777 63.8824115
12 9.0000000 35.4751414 -26.4751414 19.0602616 51.8900211
13 18.0000000 39.8879258 -21.8879258 23.9159209 55.8599306
14 25.0000000 44.3007102 -19.3007102 28.7447619 59.8566584
15 62.0000000 61.9518478 0.0481522 47.7470198 76.1566758
16 26.0000000 53.1262790 -27.1262790 38.3130930 67.9394650
17 88.0000000 92.8413386 -4.8413386 79.4769408 106.2057365
18 56.0000000 57.5390634 -1.5390634 43.0480025 72.0301243
19 11.0000000 34.5925845 -23.5925845 18.0860992 51.0990698
20 66.0000000 75.1902010 -9.1902010 61.6074334 88.7729686
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.
.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions Example Mean
Interval Intervals for Y Example Mean
Interval Plots
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Intervals for Y
The past interval was for the mean of Y given X .
What about an interval for a single observation?
So our new parameter is Yh
Yh N(0 + 1 Xh, 2
(
1 +1
n+
(Xh x)2
(x x)2
)
)
distributed t(n 2), so CI is:
Yh t(n 2, /2) s(pred)
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions Example Mean
Interval Intervals for Y Example Mean
Interval Plots
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
From our example...
Want to build a 95% CI for predicted value for an observationof 30 geese.
First, we need to find Y for X = 30.
Y30 = 26.65 + 0.88(30) = 48.71
Next, we find our standard deviation:
s(Yh) = . . .
Then we get tired and use:proc glm data=geese.geese;
model photo=observer1 / cli;
run;
Lecture 14 Psychology 790
From our example...The SAS System 00:05 Wednesday, October 25, 2006 17
The GLM Procedure
95% Confidence Limits for
Observation Observed Predicted Residual Individual Predicted Value
1 56.0000000 70.7774166 -14.7774166 -19.8244321 161.3792653
2 38.0000000 48.7134946 -10.7134946 -42.1147142 139.5417034
3 25.0000000 53.1262790 -28.1262790 -37.6431980 143.8957560
4 48.0000000 57.5390634 -9.5390634 -33.1784009 148.2565277
5 38.0000000 48.7134946 -10.7134946 -42.1147142 139.5417034
6 22.0000000 44.3007102 -22.3007102 -46.5929365 135.1943568
7 22.0000000 37.2402551 -15.2402551 -53.7720040 128.2525142
8 42.0000000 56.6565065 -14.6565065 -34.0708222 147.3838352
9 34.0000000 44.3007102 -10.3007102 -46.5929365 135.1943568
10 14.0000000 35.4751414 -21.4751414 -55.5694398 126.5197225
11 30.0000000 48.7134946 -18.7134946 -42.1147142 139.5417034
12 9.0000000 35.4751414 -26.4751414 -55.5694398 126.5197225
13 18.0000000 39.8879258 -21.8879258 -51.0778503 130.8537018
14 25.0000000 44.3007102 -19.3007102 -46.5929365 135.1943568
15 62.0000000 61.9518478 0.0481522 -28.7203344 152.6240300
16 26.0000000 53.1262790 -27.1262790 -37.6431980 143.8957560
17 88.0000000 92.8413386 -4.8413386 2.2970146 183.3856626
18 56.0000000 57.5390634 -1.5390634 -33.1784009 148.2565277
19 11.0000000 34.5925845 -23.5925845 -56.4685573 125.6537262
20 66.0000000 75.1902010 -9.1902010 -15.3866120 165.7670140
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Lecture 14 Psychology 790
Plots of Intervals
Lecture 14 Psychology 790
ANOVA table
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Whats up with that ANOVA table?
As you notice on all your regression output, you get theseANOVA tables.
The ANOVA table just refers to a partitioning of the variance.
ANOVA = Analysis of Variance
We have a total amount of error or variance present in ourdata.
What the ANOVA table does is to tell us which portion of thevariance can be accounted for by our model and whatportion is just random error.
Just as a theoretical note, we want to capture as much erroror variance by our model as possible. This means that weare accounting for changes in our data by our model.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
ANOVA table
The table shows the partitioning of the total variance, SSTO,into its two parts: sum of squares regression, SSR, and sumof squares error, SSE
Formally:
Yi Y = Yi Y (SSR) + Yi Yi(SSE)
We then also partition the degrees of freedom. We have atotal df of n 1, we have a SSR df of 1 and a SSE df of n 2(just like in our test statistic for .
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Lets go over the ANOVA tableThe GLM Procedure
Dependent Variable: photo photo count
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 1 254769.4600 254769.4600 129.20
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up
Lecture 14 Psychology 790
Our new test for 1
We start in the same way as before:
H0 : 1 = 0
Ha : 1 = 0
= 0.05
Test statistic This is where things change. Instead of ourt-test, we can compute an F test using our ANOVA table.
F (1, 43) = 129.20, p < 0.0001
Then the rest is the same:
Reject Null
We can conclude that there is a significant linear associationbetween the observer count and the true number of geese.
Lecture 14 Psychology 790
Descriptive Measures in Regression
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures Correlation Squared
Correlation Coefficient of
Determination
Wrapping Up
Lecture 14 Psychology 790
Correlation - r
The correlation is the measure of association between thetwo variables.
It is very much related to the slope in a regression model
If you recall, the equation for 1 can be expressed in termsof r.
It ranges from -1 to +1, with the extremes indicating perfectassociation.
If r = 0 there is no linear relationship between X and Y .
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures Correlation Squared
Correlation Coefficient of
Determination
Wrapping Up
Lecture 14 Psychology 790
Squared Correlation
As you can guess, r2 is not only computed for you, but it isalso the square of the correlation coefficient, r.
It is computed in terms of our ANOVA table as follows:
r2 =SSR
SSTO
Ok, great, now what does it mean conceptually?
It is a measure of the proportion of variance accounted for byour model
So, I mentioned earlier that we want to account for as muchtotal variance as possible. This lets us know how much.
In our example, we had an r2 = 0.750. This means that 75%of the total variance in total number of geese is accounted forby the observed number of geese.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures Correlation Squared
Correlation Coefficient of
Determination
Wrapping Up
Lecture 14 Psychology 790
Coefficient of Determination
Our r2 measure is not an all-powerful, all-knowing thing. Itdoes have its limitations:
A high r2 does not mean that useful predictions can bemade by the model: We may still have a high errorvariance and our predictions bands may be large.
A high r2 does not indicate that the linear line is a "goodfit": You may get a high r2, but the data is curvilinear innature, so your linear model is not appropriate.
A low r2 means X and Y are not related: Theirrelationship just might not be linear.
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up Final Thought Next Class
Lecture 14 Psychology 790
Final Thought
Hypothesis testing inregression is somethingthat is often used, andbecomes fairly routine.
The use of confidenceintervals for parameters isalso fairly straightforward.
Perhaps the most informative portion of the course fromtoday was the ideas of mean and individual predictionintervals.
These intervals are important in considering where newobservations will fall - prediction.
get_video.mpgMedia File (video/mpeg)
Overview
IntroductoryExample
Building on ourModel
Parameter Testing
Interval Estimationof Predictions
ANOVA table
DescriptiveMeasures
Wrapping Up Final Thought Next Class
Lecture 14 Psychology 790
Next Time
Kutner Chapter 3.
Regression diagnostics.
Checking assumptions.
Worrying about data.
Practical statistics.
OverviewToday's LectureOur New Schedule
Introductory ExampleSnow GeeseSnow GeeseHudson BayRegression Analysis
Building on our ModelThe ModelLets talk about Our First AssumptionOur First AssumptionLet's Put Our Assumption to Good Use
Parameter TestingParameter TestingParameter Testing - Test Statistic Parameter Testing - 5 Step ProcessGeese ExampleHypothesis Test Confidence IntervalCI in SASWhat about poor old 0? Let's Test 0 AnywayHypothesis Test - From Example Again A Note when making Inferences on s
Interval Estimation of PredictionsThe distribution of E(Y) From our example...From our example...Intervals for "705EYFrom our example...From our example...Plots of Intervals
ANOVA tableWhat's up with that ANOVA table?ANOVA table Let's go over the ANOVA table Our new test for 1
Descriptive MeasuresCorrelation - r Squared CorrelationCoefficient of Determination
Wrapping UpFinal ThoughtNext Time
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