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Last week
Discussed the ideas behind: Hypothesis testing Random Sampling Error Statistical Significance, Alpha, and p-values
Examined Correlation – specifically Pearson’s r What it’s used for, when to use it (and not to use
it) Statistical Assumptions Interpretation of r (direction/magnitude) and p
Tonight
Extend our discussion on correlation – into simple linear regression Correlation and regression are specifically
linked together, conceptually and mathematically Often see correlations paired with regression
Regression is nothing but one step past r You’ve all done it in high school math
First…brief review…
Quick Review/Quiz
A health researcher plans to determine if there is an association between physical activity and body composition. Specifically, the researcher thinks that
people who are more physically active (PA) will have a lower percent body fat (%BF).
Write out a null and alternative hypothesis
PA and %BF HO:
There is no association between PA and %BF
HA: People with ↑ PA will have ↓ %BF
The researcher will use a Pearson correlation to determine this association. He sets alpha ≤ 0.05.
Write out what that means (alpha ≤ 0.05)
Alpha If the researcher sets alpha ≤ 0.05, this
means that he/she will reject the null hypothesis if the p-value of the correlation is equal to or less than 0.05. This is the level of confidence/risk the researcher
is willing to accept
If the p-value of the test is greater than 0.05, there is a greater than 5% chance that the result could be due to ___________________, rather than a real effect/association
Results The researcher runs the correlation in SPSS and
this is in the output: n = 100, r = -0.75, p = 0.02
1) What is the direction of the correlation? What does this mean?
2) What is the sample size? 3) Describe the magnitude of the association? 4) Is this result statistically significant? 5) Did he/she fail to reject the null hypothesis OR
reject the null hypothesis?
Results defined
There is a negative, moderate-to-strong, relationship between PA and %BF (r = -0.75, p = 0.02). Those with higher levels of physical activity
tended to have lower %BF (or vice versa) Reject the null hypothesis and accept the
alternative
Based on this correlation alone, does PA cause %BF to change? Why or why not?
Error
Assume the association seen here between PA and %BF is REAL (not due to RSE). What type of error is made if the researcher fails to
reject the null hypothesis (and accepts HO) Says there is no association when there really is Type II Error
Assume the association seen here between PA and %BF is due to RSE (not REAL). What type of error is made if the researcher rejects
the null hypothesis (and rejects HO) Says there is an association when there really is not Type I Error
Our Decision
Reject HO Accept HO
What is True
HO Type I Error Correct
HA Correct Type II Error
HA: Is an association between PA and %BF HO: Is not an association between PA and %BF
Questions…?
Back to correlations
Recall, correlations provide two critical pieces of information a relationship between two variables: 1) Direction (+ or -) 2) Strength/Magnitude
However, the correlation coefficient (r) can also be used to describe how well a variable can be used for prediction (of the another). A frequent goal of statistics For example…
Association vs Prediction
Is undergrad GPA associated with grad school GPA? Can grad school GPA be predicted by undergrad GPA?
Are skinfolds measurements associated with %BF? Can %BF be predicted by skinfolds?
Is muscular strength associated with injury risk? Can muscular strength be predictive of injury risk?
Is event attendance associated with ticket price? Can event attendance be predicted by ticket price? (i.e., what ticket price will maximize profits?)
Correlation and Prediction
This idea should seem reasonable. Look at the three correlations below. In which of the
three do you think it would be easiest (most accurate) to predict the y variable from the x variable?
A B C
Correlation and Prediction
The stronger the relationship between two variables, the more accurately you can use information from one of those variables to predict the other
Which do you think you could predict more accurately?
Bench press repetitions from body weight ?
Or
40-yard dash from 10-yard dash?
Explained Variance
The stronger the relationship between two variables, the more accurately you can use information from one of those variables to predict the other
This concept is “explained variance” or “variance accounted for” Variance = the spread of the data around the center
Why the values are different for everyone Calculated by squaring the correlation coefficient, r2
Above correlation: r = 0.624 and r2 = 0.389 aka, Coefficient of Determination
What percentage of the variability in x is explained by y The 10-yard dash explains 39% of the variance in the 40-yard
dash If we could explain 100% of the variance – we’d be able to
make a perfect prediction
Coefficient of Determination, r2
What percentage of the variability in y is explained by x The 10-yard dash explains 39% of the variance in the 40-yard
dash So – about 61% (100% - 39% = 61%) of the variance remains
unexplained (is due to other things) The more variance you can explain the better the predication The less variance that is explained the more error in the
prediction Examples, notice how quickly the prediction degrades:
r = 1.00; r2 = 100% r = 0.87; r2 = 75% r = 0.71; r2 = 50% r = 0.50; r2 = 25% r = 0.22; r2 = 5%
Example with BP…
Average systolic blood pressure in the United States
Note mean – and variation (variance) in the values
Mean = 119 mmHg
SD = 20N = 22,270
Variance: BP
Why are these values so spread out?
What things influence blood pressure Age Gender Physical
Activity Diet Stress
Which of these variables do you think is most important? Least important?
If we could measure all of these, could we perfectly predict blood pressure?
Correlating each variable with BP would allow us to answer these questions using r2
Beyond r2
Obviously you want to have an estimate of how well a prediction might work – but it does not tell you how to make that prediction For that we use some form of regression
Regression is a generic term (like correlation) There are several different methods to create a
prediction equation: Simple Linear Regression Multiple Linear Regression Logistic Regression (pregnancy test) and many more…
Example using Height to predict Weight
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Height
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r = 0.81
Note the correlation coefficient above (r2 = 0.66)
SPSS is going to do all the work. It will use a process called: Least Squares Estimation
Let’s start with a scatterplot between the two variables…
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Least squares estimation: Fancy process where SPSS draws every possible line through the points - until finding the line where the vertical deviations from that line are the smallest
The green line indicates a possible line, the blue arrows indicate the deviations – longer arrows = bigger deviations
This is a crappy attempt – it will keep trying new lines until it finds the best one
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Eventually, SPSS will get it right, finding the line that minimizes deviations, known as:
Line of Best Fit
Least squares estimation: Fancy process where SPSS draws every possible line through the points - until finding the line where the vertical deviations from that line are the smallest
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The Line of Best fit is the end-product of regression
Up so many units
In so many others
And it will have a value on the y-axis for the zero value of the x-axis
-234
SLOPE
INTERCEPT
This line will have a certain slope…
The intercept can be seen more clearly if we redraw the graph with appropriate axes…
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0
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0 20 40 60 80
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-234lbs
The intercept will sometimes be a nonsense value – in this case, nobody is 0 inches tall or weighs -234 lbs.
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From the line (it’s equation), we can predict that an increase in height of 1 inch predicts a rise in weight of 5.4 lbs
We can now estimate weight from height. A person that’s 68 inches tall should weight about 135 lbs.
68
135lbs
Slope = 5.4
SPSS will output the equation, among a number of other items if you ask for them
Coefficientsa
-234.681 71.552 -3.280 .005
5.434 1.067 .806 5.092 .000
(Constant)
Height (in inches)
Model1
B Std. Error
UnstandardizedCoefficients
Beta
Standardized
Coefficients
t Sig.
Dependent Variable: Weight (in pounds)a.
SPSS output:
SLOPEINTERCEPT
The β-coefficient is the Slope of the lineThe (Constant) is the Intercept of the lineThe p-value is still here. In this case, height is a
statistically significant predictor of weight (association likely NOT due to RSE)
Depending on your high school math teacher:
Y = a + bX
SLOPEINTERCEPT
Weight = -234 + 5.434 (Height)
We can use those two values to write out the equation for our line
Y = b + mXor
Model Fit?
Once you create your regression equation, this equation is called the ‘model’ i.e., we just modeled (created a model for) the
association between height and weight
How good is the model? How well do the data fit? Can use r2 for a general comparison
How well one variable can predict the other Lower r2 means less variance accounted for, more error Our r = 0.81 for height/weight, so r2 = 0.65
We can also use Standard Error of the Estimate
How good, generally, is the fit?
Standard error of the estimate (SEE) Imagine we used our prediction equation to predict
height for each subject in our dataset (X to predict Y) Will our equation perfectly estimate each Y from X?
Unless r2 = 1.0, there will be some error between the real Y and the predicted Y
The SEE is the standard deviation of those differences The standard deviation of actual Y’s about predicted Y’s Estimates typical size of the error in predicting Y (sort of)
Critically related to r2, but SEE is more specific to your equation
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Let’s go back to our line of best fit (this line represents the predicted value of Y for each X):
Notice some real Y’s are closer to the line than others
SEE = The standard deviation of actual Y’s about predicted Y’s
Large Error
Small Error
Very Small Error
SEE is the standard deviation of all these errors
SEE Why calculate the ‘standard deviation’ of these errors
instead of just calculating the ‘average error’? By using standard deviation instead of the mean, we can
describe what percentage of estimates are within 1 SEE of the line In other words, if we used this prediction equation, we would expect that
68% fall within 1 SEE 95% fall within 2 SEE 99% fall within 3 SEE
Knowing, “How often is this accurate?” is probably more important than asking, “What’s the average error?”
Of course, how large the SEE is depends on your r2 and your sample size (larger samples make more accurate predictions)
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Let’s go back to our line of best fit :
In regression, we call these errors/deviations “residuals”
Residual Y = Real Y – Predicted Y
Notice that some of the residuals are - and some are +, where we over-estimated (-) or under-estimated (+) weight
Large Residual
Small Residual
Very Small Residual
SEE is the standard deviation of the residuals
Residuals
The line of best fit is a line where the residuals are minimized (least error) The residuals will sum to 0 The mean of the residuals will also be 0 The Line of Best Fit is the ‘balance point’ of the
scatterplot The standard deviation of the residuals is the SEE
Recognize this concept/terminology– if there is a residual – that means the effect of other variables is creating error Confounding variables create residuals
QUESTIONS…?
Statistical Assumptions of Simple Linear Regression See last week’s notes on assumptions of
correlation… Variables are normally distributed Homoscedasticity of variance Sample is representative of population Relationship is linear (remember, y = a + bX) The variables are ratio/interval (continuous)
Can’t use nominal or ordinal variables …at least pretend for now, we’ll break this one
next week.
Simple Linear Regression: Example Let’s start simple, with two variables we
know to be very highly correlated 40-yard dash and 20-yard dash
Can we predict 40-yard dash from 20-yard dash?
Correlation
Strength? Direction? Statistically significant correlations will (usually)
produce statistically significant predictors r2 = ??
0.66
Now, run the regression in SPSS
Model Outputs
Adjusted r2 = Adjusts the r2 value based on sample size…small samples tend to overestimate the ability to predict the DV with the IV (our sample is 428, adjusted is similar)
Model Outputs
Notice our SEE of 0.06 seconds. 68% of residuals are within 0.06 seconds of predicted 95% of residuals are within 0.12 seconds of predicted
Model Outputs
The ‘ANOVA’ portion of the output tells you if the entire model is statistically significant. However, since our model just includes one variable (20-yard dash), the p-value here will match the one to follow
Outputs
Y-intercept = 1.259 Slope = 1.245 20-yard dash is a statistically significant predictor What is our equation to predict 40-yard dash?
Equation 40yard dash time =
1.245(20yard time) + 1.259 If a player ran the 20-yard dash in 2.5 seconds,
what is their estimated 40-yard dash time?1.245(2.5) + 1.259 =
4.37 secondsIf the player actually ran 4.53 seconds, what is
the residual?Residual = Real – Predicted
4.53 – 4.37 = 0.16
Significance vs. Importance in Regression A statistically significant model/variable does NOT
mean the equation is good at predicting
The p-value tells you if the independent variable (predictor) can be used as a predictor of the dependent variable
The r2 tells you how good the independent variable might be as a predictor (variance accounted for)
The SEE tells you how good the predictor (model) is at predicting
QUESTIONS…?