Plan for today

• Hypothesis Testing
• Difference in Means
• Regressions

Why do we need hypothesis testing?

• We do not know true population parameters (only the sample)
• By using what we observe we can approximate the true population

• Either the two groups have different means (or the same)
• Either Aly has a higher mean than Bill (or Bill \(\geq\) Aly)
• Either Bill has a higher mean than Aly (or Aly \(\geq\) Bill)
• All three of these are examples of testable hypotheses!

• In hypothesis testing, there is always 2 hypotheses
• \(H_0:\) The Null Hypothesis
• \(H_a:\) The Alternative Hypothesis
• The Null Hypothesis is what you are actually testing
• The Alternative Hypothesis is the other possible situation
• Typically the Null Hypothesis is assumed true, until proven false

• Let's use the first example from the last slide:

• \(H_0: mean(Aly) = mean(Bill)\)
• \(H_a: mean(Aly) \ne mean(Bill)\)

• If we fail to reject our null hypothesis we say:
• "We failed to reject null hypothesis"
• NOT "We proved the alternative hypothesis"

• Using Probability Theory to Measure Type 1 Errors

• Sample vs Population

Formula for differences in means:

Checking if a mean of one group is greater than a value (V):

• Since this is greater than 1.96, we say there is a significant difference with at least 95% Confidence!

• This is more simply done with the t.test function:

t.test(A, B)

## 
##  Welch Two Sample t-test
## 
## data:  A and B
## t = 5.2475, df = 57.047, p-value = 2.358e-06
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  3.348931 7.481990
## sample estimates:
## mean of x mean of y 
##  49.88979  44.47433

• Looking at the Distribution

• Two different kinds of hypothesis tests
• \(H_0\) that contain "=" signs or those with ">" and "<"

• The two populations (not samples) are normally distributed
• Each value is sampled independently from each other value. This assumption requires that each subject provide only one value. If a subject provides two scores, then the scores are not independent.

• The first assumption, if violated, increases the chances of rejecting the null hypothesis when it is true
• The second assumption, if violated, severely undermines the integrity of the test
• If not normally distributed assign a penalty to your difference test
• If samples are not independent, aggregate by individual

• We want to know IF there is a relationship between X and Y
• What DIRECTION is the relationship?
• What is the MAGNITUDE of the relationship?
• Is the relationship STATISTICALLY SIGNIFICANT?

• Perhaps we want to know if urban communities are more at risk for Cirrhosis
• The scatter-plot seems to indicate a positive correlation

• What other variables might be responsible?
• Liquor consumption
• Wine consumption
• Let's try adding them to the model

• Add an error term, and that is a regression model!

Although for each variable in the regression we increase dimensions, we can still look at a cross section of our plot.

• Minimizes square distance between line and observation

## 
## Call:
## lm(formula = Cirrhosis_death_rate ~ Pct_urban + Liquor_consumption_per_capita + 
##     Wine_consumption_per_capita, data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -28.5939  -5.0002   0.7397   7.2051  18.1331 
## 
## Coefficients:
##                               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                     3.8706     7.1618   0.540 0.591738    
## Pct_urban                       0.4965     0.1414   3.512 0.001078 ** 
## Liquor_consumption_per_capita   0.2286     0.1002   2.281 0.027702 *  
## Wine_consumption_per_capita     1.6008     0.3919   4.085 0.000194 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.96 on 42 degrees of freedom
## Multiple R-squared:  0.796,  Adjusted R-squared:  0.7814 
## F-statistic: 54.62 on 3 and 42 DF,  p-value: 1.503e-14

• How well does our model explain the variance in Y (Cirrhosis Death Rate)?

summary(mod)$r.squared

## [1] 0.7959899

summary(mod)$adj.r.squared

## [1] 0.7814178

• Other models may be better for prediction
• linear model better for interpretation of relationship
• Don't have to worry as much about over-fitting

• Direction of causality
• Omitted variable bias
• Linearity of Parameters
• Conditional Expected Value of Error is 0
• No Heteroskedasticity
• No Perfect Co-linearity
• Data is Randomly 'Drawn' from the Population

Direction of Causality

• Need to know if X causes Y
• Or if Y causes X

Ommitted variable bias

• Is there something else that is related to our dependent variable not in the model?
• If so, it can skew the coefficients and our results

Linearity of Parameters

• Are the coefficients linear?
• Could there be a log or polynomial relationship?

Conditional Expected Value of Error is 0

• Error is centered around 0
• Is violated with omitted variables
• Improperly transformed variable can skew the error as well

No Heteroskedasticity

No Perfect Co-linearity

##    male female
## 1:    1      0
## 2:    0      1
## 3:    1      0
## 4:    0      1
## 5:    0      1

Data is Randomly 'Drawn' from the Population

• No selection bias
• No multiple observations from one individual