1. Unit 4: Inferences about a Single Quantitative Predictor

2. Unit Organization
First consider details of simplest model (one parameter estimate; mean-only model; no X’s)
Next examine simple regression (two parameter estimates, one X for one quantitative predictor variable)
These provide critical foundation for all linear models
Subsequent units will generalize to one dichotomous predictor variable (Unit 5; Markus), multiple predictor variables (Units 6-7) and beyond….

3. Linear Models as Models
Linear models (including regression) are ‘models’
DATA = MODEL + ERROR
Three general uses for models:
Describe and summarize DATA (Ys) in a simpler form using MODEL
Predict DATA (Ys) from MODEL
Will want to know precision of prediction. How big is error? Better prediction with less error.
Understand (test inferences about) complex relationships between individual regressors (Xs) in MODEL and the DATA (Ys). How precise are estimates of relationship?
MODELS are simplifications of reality. As such, there is ERROR. They also make assumptions that must be evaluated

4. Fear Potentiated Startle (FPS)
We are interested in producing anxiety in the laboratory To do this, we develop a procedure where we expose people to periods of unpredictable electric shock administration alternating with periods of safety. We measure their startle response in the shock and safe periods. We use the difference between their startle during shock – safe to determine if they are anxious. This is called Fear potentiated startle (FPS). Our procedure works if FPS > 0. We need a model of FPS scores to determine if FPS > 0.

5. Fear Potentiated Startle: One parameter model
A very simple model for the population of FPS scores would predict the same value for everyone in the population.
Yi = 0
We would like this value to be the “best” prediction.
In the context of DATA = MODEL + ERROR, how can we quantify “best”? ^
We want to predict some characteristic about the population of FPS scores that minimizes the ERROR from our model.
ERROR = DATA – MODEL
i = Yi – Yi; There is an error (i) for each population score. How can we quantify total model error? ^

6. Total Error
Sum of errors across all scores in the population isn’t ideal b/c positive and negative errors will tend to cancel each other
(Yi – Yi)
Sum of absolute value of errors could work. If we selected 0 to minimize the sum of the absolute value of errors,  would equal the median of the population.
( |Yi – Yi| )
Sum of squared errors (SSE) could work. If we selected  to minimize the sum of squared errors, 0 would equal the mean of the population. ^ ^ ^
(Yi – Yi)2

7. One parameter model for FPS
For the moment, lets assume we prefer to minimize SSE (more on that in a moment). You should predict the population mean FPS for everyone.
Yi = 0 where  0 = 
What is the problem with this model and how can we fix this problem? ^
We don’t know the population mean for FPS scores ().
We can collect a sample from the population and use the sample mean (X) as an estimate of the population mean (). X is an unbiased estimate for 

8. Model Parameter Estimation
Population model
Yi = 0 where  0 = 
Yi = 0 + i
Estimate population parameters from sample
Yi = b0 where b0 = X
Yi = b0 + ei ^ ^

9. Least Squares Criterion
In ordinary least squares (OLS) regression and other least squares linear models, the model parameter estimates (e.g., b0) are calculated such that they minimize the sum of squared errors (SSE) in the sample in which you estimate the model.
SSE =  (Yi – Yi)2
SSE = ei2 ^

10. Properties of Parameter Estimates
There are 3 properties that make a parameter estimate attractive.
Unbiased: Mean of the sampling distribution for the parameter is equal to the value for that parameter in the population.
Efficient: The sample estimates are close to the population parameter. In other words, the narrower the sampling distribution for any specific sample size N, the more efficient the estimator. Efficient means small SE for parameter estimate
Consistent: As the sample size increases, the sampling distribution becomes narrower (more efficient). Consistent means as N increases, SE for parameter estimate decreases

11. Least Squares Criterion
If the i are normally distributed, both the median and the mean are unbiased and consistent estimators.
The variance of the sampling distribution for the mean is: 2 N
The variance of the sampling distribution for the median is:
2 2N
Therefore the mean is the more efficient parameter estimate.
For this reason, we tend to prefer to estimate our models by minimizing the sum of squared errors.

12. Fear-potentiated startle during Threat of Shock
> setwd("C:/Users/LocalUser/Desktop/GLM")
> d = lm.readDat('4_SingleQuantitative_FPS.dat')
> str(d)
'data.frame': 96 obs. of 2 variables:
$ BAC: num
$ FPS: num
> head(d)
BAC FPS
> some(d)
BAC FPS

13. Descriptives and Univariate Plots
> lm.describeData(d)
var n mean sd median min max skew kurtosis
BAC
FPS
> windows() #on MAC, use quartz()
> par('cex' = 1.5, 'lwd'=2)
> hist(d$FPS)

14. FPS Experiment: The Inference Details
Goal: Determine if our shock threat procedure is effective at potentiating startle (increasing startle during threat relative to safe) Create a simple model of FPS scores in the population FPS = 0 Collect sample of N=96 to estimate 0 Calculate sample parameter estimate (b0) that minimizes SSE in sample Use b0 to test hypotheses H0: 0 = 0 Ha: 0 <> 0

15. Estimating a one parameter model in R
m = lm(FPS ~ 1, data = d)
> summary(m)
Call:
lm(formula = FPS ~ 1, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-13 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 95 degrees of freedom

16. Errors/Residuals This is simple descriptive information about the
summary(m)
Call:
lm(formula = FPS ~ 1, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-13 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 95 degrees of freedom
This is simple descriptive information about the
errors (ei)
ei = (Yi – Yi) ^

17. Errors/Residuals ^ ei = (Yi – Yi)
R can report errors for each individual in the sample:
> residuals(m)
You can get the SSE easily:
> sum(residuals(m)^2)
[1]

18. Standard Error of Estimate
summary(m)
Call:
lm(formula = FPS ~ 1, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-13 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 95 degrees of freedom
This is the standard error of estimate. It an estimate of the standard deviation of i
 (Yi – Yi)2 SSE
N - P N - P
NOTE: for mean-only model, this is sY ^

19. Coefficients (Parameter Estimates)
summary(m)
Call:
lm(formula = FPS ~ 1, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-13 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 95 degrees of freedom
This is b0, the unbiased sample estimate of 0, and its standard error. It is also called the intercept in regression (more on this later). Yi = b0 Yi = 32.2
> coef(m)
(Intercept) ^

20. Predicted Values ^
Yi = 32.19
You can get the predicted value for each individual in the sample using this model:
> fitted.values(m)

21. Testing Inferences about 0
summary(m)
Call:
lm(formula = FPS ~ 1, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-13 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 95 degrees of freedom
This is the t-statistic to test the H0 that 0 = 0.
The probability (p-value) of obtaining a sample b0 = 32.2 if H0 is true (0 = 0) <
Describe the logic of how this was determined?

22. Sampling Distribution: Testing Inferences about 0
H0: 0 = 0; Ha: 0 <> 0
If H0 is true, the sampling distribution for 0 will have a mean of 0. We can estimate standard deviation of the sampling distribution with SE for b0.
t (df=N-P) = b0 – = – 0 = 8.40
SEb
b0 is approximately 8 standard deviations above the expected mean of the distribution if H0 is true
pt(8.40,95,lower.tail=FALSE) * 2
[1] e-13
The probability of obtaining a sample
b0 = 32.2 if H0 is true is very
low (< .05). Therefore we reject H0
And conclude that 0 <> 0 and b0 is
our best (unbiased) estimate of it.

23. Statistical Inference and Model Comparisons
Statistical inference about parameters is fundamentally about model comparisons You are implicitly (t-test of parameter estimate) or explicitly (F-test of model comparison) comparing two different models of your data We follow Judd et al and call these two models the compact model and the augmented model. The compact model will represent reality as the null hypothesis predicts. The augmented model will represent reality as the alternative hypothesis predicts. The compact model is simpler (fewer parameters) than the augmented model. It is also nested in the augmented model (i.e. a subset of parameters)

24. Model Comparisons: Testing inferences about 0^
FPSi = 0 H0: 0 = 0 Ha: 0 <> 0 Compact model: FPSi = 0; Augmented model: FPSi = 0 ( b0) We estimate 0 parameters (P=0) in this compact model We estimate 1 parameter (P=1) in this augmented model Choosing between these two models is equivalent to testing if 0 = 0 as you did with the t-test ^ ^

25. Model Comparisons: Testing inferences about 0^
Compact model: FPSi = 0
Augmented model: FPSi = 0 ( b0)
We can compare (and choose between) these two models by comparing their total error (SSE) in our sample
SSE = (Yi – Yi)2
SSE(C) = (Yi – Yi)2 = (Yi – 0)2
> sum((d$FPS - 0)^2)
[1]
SSE(A) = (Yi – Yi)2 = (Yi – 32.19)2
> sum((d$FPS – coef(m)[1])^2 #(sum(residuals(m)^2)
[1] ^ ^ ^

26. Model Comparisons: Testing inferences about 0^
Compact model: FPSi = 0;
SSE = 233,368.3
P = 0
Augmented model: FPSi = 0 ( b0)
SSE = 133,888.3
P=1
F (PA – PC, N – PA) = (SSE(C) -SSE(A)) / (PA-PC)
SSE(A) / (N-PA)
F (1– 0, 96 – 1) = ( ) / (1 - 0)
/ (96 - 1)
F(1,95) = 70.59, p < .0001
> pf( ,1,95, lower.tail=FALSE)
[1] e-13 ^

27. Effect Sizes
Your parameter estimates are descriptive. The describe effects in the original units of the (IVs) and DV. Report them in your paper There are many other effect size estimates available. You will learn two that prefer. Partial eta2 (p2): Judd et al call this PRE (proportional reduction in error) Eta2 (2): This is also commonly referred to as R2 in regression.

28. Sampling Distribution vs. Model Comparison
The two approaches to testing H0 about parameters (0, j) are statistically equivalent
They are complementary approaches with respect to conceptual understanding of GLMs
Sampling distribution
Focus on population parameters and their estimates
Tight connection to sampling and probability distributions
Understanding of SE (sampling error/power; confidence intervals; graphic displays)
Model comparison
Focus on models themselves increase
Highlights model fit (SSE) and model parsimony (P)
Clearer link to PRE (p2)
Test comparisons that differ by > 1 parameter (discouraged)

29. Partial Eta2 or PRE ^ Compact model: FPSi = 0; SSE = 233,368.3 P = 0
Augmented model: FPSi = 0 ( b0)
SSE = 133,888.3
P=1
How much was the error reduced in the augmented model relative to the compact model?
SSE(C) – SSE(A) = 233, ,888.3 = .426
SSE (C) ,368.3
Our more complex model that includes 0 reduces prediction error (SSE) by approximately 43%. Not bad! ^

30. Confidence Interval for b0
A confidence interval (CI) is an interval for a parameter estimate in which you can be fairly confident that you will capture the true population parameter (in this case, 0). Most commonly reported is the 95% CI. Across repeated samples, 95% of the calculated CIs will include the population parameter*.
> confint(m)
2.5 % %
(Intercept)
Given what you now know about confidence intervals and sampling distributions, what should the formula be?
CI (b0) = b0 + t (; N-P) * SEb0
For the 95% confidence interval this is approximately + 2 SEs around our unbiased estimate of 0

31. Confidence Interval for b0
How can we tell if a parameter is “significant” from the confidence interval?
If a parameter <> 0 at  = .05, then the 95% confidence interval for its parameter estimate should not include 0.
This is also true for testing whether the parameter is equal to any other True for any other non-zero point estimate for the parameter non-zero value

32. The one parameter (mean-only) model: Special Case
What special case (specific analytic test) is statistically equivalent to the test of the null hypothesis: 0 = 0 in the one parameter model?
The one sample t-test testing if a population mean = 0.
> t.test(d$FPS)
One Sample t-test
data: d$FPS
t = , df = 95, p-value = 4.261e-13
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x

33. Testing 0 = non-zero values
How could you test an H0 regarding 0 = some value other than 0 (e.g., 10)? HINT: There are at least three methods.
Option 1: Compare SSE for the augmented model (Yi = 0 ) to SSE from a different compact model for this new H0 (Yi = 10)
Option 3: Does the confidence interval for the parameter estimate contain this other value? No p-value provided.
> confint(m)
2.5 % %
(Intercept)
Option 2: Recalculate t-statistic using this new H0.
t = b0 – 10
SEb0 ^

34. Intermission….. One parameter (0) “mean-only” model
Description: b0 describes mean of Y
Prediction: b0 is predicted value that minimizes sample SSE
Inference: Use b0 to test if 0 = 0 (default) or any other value. One sample t-test.
Two parameter (0, 1) model
Description: b1 describes how Y changes as function of X1. b0 describes expected value of Y at specific value (0) for X1. Prediction: b0 and b1 yield predicted values that vary by X1 and minimize SSE in sample.
Inference: Test if 1 = 0. Pearson’s r; independent sample t-test. Test if 0 = 0. Analogous to one-sample t-test controlling for X1, if X1 is mean-centered. Very flexible!

35. Two Parameter (One Predictor) models
We started with a very simple model of FPS: FPS = 0
What if some participants were drunk and we knew their blood alcohol concentrations (BAC). Would it help? What would the model look like? What question (s) does this model allow us to test? Think about it

36. The Two Parameter Model
DATA = MODEL + ERROR
Yi = 0 + 1 * X1 + i
Yi = 0 + 1 * X1
i = Yi - Yi
FPSi = 0 + 1 * BAC1 ^ ^ ^

37. The Two Parameter Model^
Yi = 0 + 1 * X1
As before, the population parameters in the model (0 , 1) are estimated by b0 & b1 calculated from sample data based on the least squares criterion such that they minimize SSE in the sample data.
Sample model
Yi = b0 + b1 * X1 ^
To derive these parameter estimates you must solve series of simultaneous equations using linear algebra and matrices (see supplemental reading).
Or use R!

38. Least Squares Criterion^
ei = Yi – Yi
SSE = ei2

39. Interpretation of b0 in Two Parameter Model^
Yi = b0 + b1 * X1
b0 is predicted value for Y when X1 = 0. Graphically, this is the Y intercept for the regression line (Value of Y where regression line crosses Y-axis at X1 = 0*).
Approximately what is b0 in this example?
42.5

40. Interpretation of b0 in Two Parameter Model
IMPORTANT: Notice that b0 is very different (42.5) in the two parameter model than in previous one parameter model (32.2)
WHY?
In the one parameter model b0 was our sample estimate of the mean FPS score in everyone. b0 in the two parameter model is our sample estimate of the mean FPS score for people with BAC = 0, not everyone.

41. Interpretation of b1 in Two Parameter Model^
Yi = b0 + b1 * X1
b1 is the predicted change in Y for every one unit change in X1. Graphically it is represented by the slope of the line regression line. If you understand the units of your predictor and DV, this is an attractive description of their relationship. ^
FPSi = * BACi
For every 1% increase in BAC, FPS decreases by microvolts.
For every .01% increase in BAC, FPS decreases by microvolts.

42. Testing Inferences about 1
Does alcohol affect people’s anxiety? FPSi = 0 + 1 * BACi What are your null and alternative hypotheses about model parameter to evaluate this question? ^
H0: 1 = 0
Ha: 1 <> 0
If 1 = 0, this means that FPS does not change with changes in BAC. In other words, there is no effect of BAC on FPS. If 1 < 0, this means that FPS decreases with increasing BAC (People are less anxious when drunk) If 1 > 0, this means FPS increases with increasing BAC (i.e., people are more anxious when drunk).

43. Estimating a Two Parameter Model in R
> m2 = lm(FPS ~ BAC, data = d)
> summary(m2)
Call:
lm(formula = FPS ~ BAC, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-09 ***
BAC
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 94 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 1 and 94 DF, p-value:

44. Testing Inferences about 1
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-09 ***
BAC
Does BAC affect FPS? Explain this conclusion in terms of the parameter estimate, b1 and its standard error
Under the H0: 1 = 0, the sampling distribution for 1 will have a mean of 0 with an estimated standard deviation of
t (96-1) = – 0 = -1.92
95.894
Our this value of the parameter estimate, b1, is 1.92 standard deviations below the expected mean of the sampling distribution for H0.
> pt(-1.92, 94, lower.tail = TRUE)*2
[1]
A b1 of this size is not unlikely under the null, therefore you fail to reject the null and conclude that BAC has no effect on FPS

45. Testing Inferences about 1
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-09 ***
BAC
One tailed p-value
> pt(-1.92, 94, lower.tail = TRUE)
[1]
Two-tailed p-value
> pt(-1.92, 94, lower.tail = TRUE)*2
[1]
H0: 1 = 0
Ha: 1 <> 0

46. Model Comparison: Testing Inferences about 1
H0: 1 = 0 Ha: 1 <> 0 What two models are you comparing when you test hypotheses about 1? Describe the logic.
Compact Model: FPSi = 0 + 0 * BACi
PC = 1
SSE(C) =
Augmented Model: FPSi = 0 + 1 * BACi
PA = 2
SSE(A) =
F(PA-PC, N-PA) = SSE(C) – SSE(A) / (PA-PC)
SSE(A) / (N-PA)
F(1,94) = , p = ^

47. Sum of Squared Errors
If there is a perfect relationship between X1 and Y in your sample, what will the SSE be in the two parameter model and why?
SSE(A) = 0 . All data points will fall perfectly on the regression line. All errors will be 0
If there is no relationship at all between X1 and Y in your sample (b1 = 0), what will the SSE be in the two parameter model and why?
SSE(A) = SSE of the mean-only model. X1 provides no additional information about the DV. Your best prediction will still be the mean of the DV.

48. Testing Inferences about 0
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-09 ***
BAC
What is the interpretation of b0 in this two parameter model?
It is the predicted FPS for a person with BAC = 0 (sober).
The test of this parameter estimate could inform us if the shock procedure worked among our sober participants. This is probably a more appropriate manipulation check than testing if it worked in everyone including drunk people given that alcohol could have reduced FPS.
What two models are being compared?
Compact Model: FPSi = 0 + 1 * BACi
Augmented Model: FPSi = 0 + 1 * BACi ^

49. Confidence Interval for bj or b0
You can provide confidence intervals for each parameter estimate in your model.
> confint(m2)
2.5 % 97.5 %
(Intercept)
BAC
The underlying logic from your understanding of sampling distributions remains the same
CI (b) = b + t (;N-P) * SEb where P = total # of parameters
How can we tell if a parameter is “significant” from the confidence interval?
If a parameter is <> 0, at  = .05, then the 95% confidence interval should not include 0. True for any other non-zero value for b as well.

50. Partial Eta2 or PRE for 1
How can you calculate the effect size estimate partial eta2 (PRE) for 1
Compare the SSE across the two relevant models
Compact Model: FPSi = 0 + 0 * BACi
SSE(C) =
Augmented Model: FPSi = 0 + 1 * BACi
SSE(A) =
SSE(C) – SSE(A) = =
SSE (C)
Our augmented model that includes a non-zero effect for BAC reduces prediction error (SSE) by only 3.8% over the compact model that fixes this parameter at 0. ^

51. Partial Eta2 or PRE for 0
How can you calculate the effect size estimate partial eta2 (PRE) for 0
Compare the SSE across the two relevant models
Compact Model: FPSi = 0 + 1 * BACi
SSE(C) =
Augmented Model: FPSi = 0 + 1 * BACi
SSE(A) =
SSE(C) – SSE(A) = = SSE (C)
Our augmented model that allows FPS to be non-zero for people with BAC=0 (sober people) reduces reduces prediction error (SSE) by 30.9% from the model that fixes FPS at 0 when BAC =0! ^

52. Coefficient of Determination (R2)
Proportion of explained variance (i.e., proportion of variance in Y accounted for by all Xs in model).
DATA = MODEL + ERROR
For individuals:
Yi = Yi + ei
With respect to variances:
sYi2 = sYi2 + sei2
R2 = sYi2
sYi2
> var(fitted.values(m2))/ var(d$FPS)
[1] ^ ^

53. Coefficient of Determination (R2)
> m2 = lm(FPS ~ BAC, data = d)
> summary(m2)
Call:
lm(formula = FPS ~ BAC, data = d)
Residuals:
Min Q Median Q Max
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) e-09 ***
BAC
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: on 94 degrees of freedom
Multiple R-squared: , Adjusted R-squared:
F-statistic: on 1 and 94 DF, p-value:

54. R2 and the Mean-Only Model
Why did the mean-only model not have an R2?
It explained no variance in Yi because it predicted the same value (mean) for every person. The variance of the predicted values is 0 in the mean-only model
In fact, the SSE for the mean-only model is the numerator of the formula for the variance for Yi
SSE = (Yi – Y)2
S2 = (Yi – Y)2
N-1 ^

55. R2 and the Mean-Only Model
The mean-only model is used in an alternative conceptualization of R2 for any augmented model
R2 = SSE(Mean-only) - SSE(A)
SSE(Mean-only)
Mean-Only Model: FPSi = 0
SSE(Mean-only) =
Augmented Model: FPSi = 0 + 1 * BACi
SSE(A) =
R2 = =
In this augmented model, R2 is fully accounted for by BAC. In more complex models, R2 will be the aggregate of multiple predictors. ^ ^

56. Test of 1 in Two Parameter Model: Special Case
When both the predictor variable and the dependent variable are quantitative, the test of 1 = 0 is statistically equivalent to the what other common statistical test
The test of the Pearson’s correlation coefficient, r
> corr.test(d$BAC, d$FPS)
Correlation matrix
[1]
Sample Size
[,1]
[1,] 96
Probability values adjusted for multiple tests.
[1,] 0.058
Furthermore r2 = R2 for this model only.
= .038

57. Visualizing the Model
e = effect('BAC', m2) plot(e)

58. Displaying Model Results
Error bars/bands
You are predicting the mean Y for any X. There is a sampling distribution around this mean. The true population mean Y for any X is uncertain. You can display this uncertainty by displaying information about the sampling distribution at any/every X. This is equivalent to error bars in ANOVA.

59. ^
Error Band for Yi
plot(d$BAC,d$FPS, xlim = c(0,.15), xlab = 'Blood Alcohol Concentration', ylab = 'Fear-potentiated startle')
dNew = data.frame(BAC= seq(0,.14,.0001))
pY = lm.pointEstimates(m2,dNew)
lines(dNew$BAC,pY[,1], col='red', lwd=2)
lines(dNew$BAC,pY[,2], col='gray', lwd=.5)
lines(dNew$BAC,pY[,3], col='gray', lwd=.5)
effect() dsiplays 95% CI for However, I prefer + 1 SE

60. Error Band for Yi ^ Why are the error bands not linear?
Model predictions are better (less error) near the center of your data (Xi). Regression line will always go through mean of X and Y. Small changes in b1 across samples will produce bigger variation in Yi at the edge of the model (far from the mean X).
FPSi = * BACi
confint(m2)
2.5 % %
(Intercept)
BAC ^

61. Error Band for Compare to the SE for b0
b0 is simply the predicted value for Y when X = 0.
We can use additive transformations of X to make tests of the predicted value at X = 0. Most common in repeated measures designs but used elsewhere as well.

62. Publication Quality Figure