Without using your book, derive expressions for \(\hat{\beta}_0\) and \(\hat{\beta}_1\) by solving for them in the normal equations:
\[ \sum_{i=1}^n y_i = n\hat{\beta}_0 + \hat{\beta}_1 \sum_{i=1}^n x_i \\ \sum_{i=1}^n x_i y_i = \hat{\beta}_0 \sum_{i=1}^n x_i + \hat{\beta}_1 \sum_{i=1}^n x_i^2 \]
note: \(\frac{1}{n}\sum_{i=1}^n x_i\) can be rewritten \(\bar{x}\).
download.file("http://www.openintro.org/stat/data/mlb11.RData",
destfile = "mlb11.RData")
load("mlb11.RData")
plot_ss(poverty$Poverty, poverty$Graduates)
The workhorse function: lm()
m1 <- lm(Graduates ~ Poverty, data = poverty)
The formula notation is read: "I'd like to express y as a function of x". It creates a rich object of class lm.
class(m1)
## [1] "lm"
names(m1)
## [1] "coefficients" "residuals" "effects" "rank" ## [5] "fitted.values" "assign" "qr" "df.residual" ## [9] "xlevels" "call" "terms" "model"
You can extract info about your model using
Attributes: reference the attributes found with names() by using the $ operator.
Summary: the most useful information can be displayed using the summary() command.
Print: print your model object to get the basic coefficient estimates.
## ## Call: ## lm(formula = Graduates ~ Poverty, data = poverty) ## ## Residuals: ## Min 1Q Median 3Q Max ## -5.954 -1.820 0.544 1.515 6.199 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 96.202 1.343 71.65 < 2e-16 *** ## Poverty -0.898 0.114 -7.86 3.1e-10 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 2.5 on 49 degrees of freedom ## Multiple R-squared: 0.558, Adjusted R-squared: 0.549 ## F-statistic: 61.8 on 1 and 49 DF, p-value: 3.11e-10
The slope, \(\beta_1\), tells you that a one unit increase in the \(x\) is associated with a \(\beta_1\) unit increase in the \(y\), on average.
A one percentage point increase in a state's poverty rate is associated with a -0.8979 decrease in the state's graduation rate, on average.
Why the "on average"?
\[ \hat{E}(Y|X) = \hat{\beta}_0 + \hat{\beta}_1 * x \]
note: sign of \(\beta_1\) is often more interesting than magnitude (effected by scaling).
Mathematically: The expected value of \(y\) when \(x\) is zero.
Contextually: Sometimes, the "start-up" value of the \(y\).
Does the slope have meaning in the poverty vs grad rate example?
Recall the last question from homework 1: would you expect a relationship between the magnitude of an earthquake and the number of stations that detect it?