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Simple Linear Regression

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Simple Linear Regression

- 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

- 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…

- Correlation and regression are specifically linked together, conceptually and mathematically

- 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

- 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)

- 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

- 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?

- 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?

- 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

- What type of error is made if the researcher fails to reject the null hypothesis (and accepts HO)
- 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

- What type of error is made if the researcher rejects the null hypothesis (and rejects HO)

- HA: Is an association between PA and %BF
- HO: Is not an association between PA and %BF

Questions…?

- 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…

- 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?)

- 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

- 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?

- 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

- Variance = the spread of the data around the center

- 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…

Mean = 119 mmHg

SD = 20

N = 22,270

- Average systolic blood pressure in the United States
- Note mean – and variation (variance) in the values

Why are these values so spread out?

- 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

- 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…

- There are several different methods to create a prediction equation:

Example using Height to predict Weight

r = 0.81

Let’s start with a scatterplot between the two variables…

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

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

r = .81

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

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

r = .81

Eventually, SPSS will get it right, finding the line that minimizes deviations, known as:

Line of Best Fit

r = .81

Up so many units

In so many others

-234

The Line of Best fit is the end-product of regression

This line will have a certain slope…

SLOPE

And it will have a value on the y-axis for the zero value of the x-axis

INTERCEPT

-234lbs

The intercept can be seen more clearly if we redraw the graph with appropriate axes…

The intercept will sometimes be a nonsense value – in this case, nobody is 0 inches tall or weighs -234 lbs.

r = .81

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

135lbs

Slope = 5.4

68

We can now estimate weight from height. A person that’s 68 inches tall should weight about 135 lbs.

SPSS will output the equation, among a number of other items if you ask for them

INTERCEPT

SLOPE

SPSS output:

The β-coefficient is the Slope of the line

The (Constant) is the Intercept of the line

The p-value is still here. In this case, height is a statistically significant predictor of weight (association likely NOT due to RSE)

SLOPE

INTERCEPT

We can use those two values to write out the equation for our line

Depending on your high school math teacher:

Y = b + mX

or

Y = a + bX

Weight = -234 + 5.434 (Height)

- 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

- Can use r2 for a general comparison

- 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

Let’s go back to our line of best fit (this line represents the predicted value of Y for each X):

r = .81

SEE is the standard deviation of all these errors

Large Error

Very Small Error

Small Error

- Notice some real Y’s are closer to the line than others
- SEE = The standard deviation of actual Y’s about predicted Y’s

- 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

- In other words, if we used this prediction equation, we would expect that
- 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)

Let’s go back to our line of best fit :

r = .81

SEE is the standard deviation of the residuals

Very Small Residual

Large Residual

Small Residual

- 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

- 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…?

- See last week’s notes on assumptions of correlation…
- Variables are normally distributed
- Homoscedasticityof 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.

- 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?

- Trimmed dataset down to just two variables
- Let’s look at a scatterplot first

All my assumptions are good, should be able to produce a decent prediction

Next step, correlation

- Strength? Direction?
- Statistically significant correlations will (usually) produce statistically significant predictors
- r2 = ??

0.66

Now, run the regression in SPSS

The ‘predictor’ is the independent variable

- 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)

- 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

- 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

- Y-intercept = 1.259
- Slope = 1.245
- 20-yard dash is a statistically significant predictor
- What is our equation to predict 40-yard dash?

- 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 seconds

If the player actually ran 4.53 seconds, what is the residual?

Residual = Real – Predicted

4.53 – 4.37 = 0.16

- 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 beas a predictor (variance accounted for)
- The SEE tells you how good the predictor (model) is at predicting

QUESTIONS…?

- In-class activity…
- Homework:
- Cronk Section 5.3
- Holcomb Exercises 29, 44, 46 and 33

- Multiple Linear Regression next week