PSY 293 Exam 2

Linear Regression:

Regression line→ line that best fits both variables

• Known as line of best fit 

• Only deals with linear relationships

• The model is a straight line and from that model we can predict the other variable

Formula→ y=bx+a

• B is slope

• A is intercept 

Linear Regression Coefficients:

Formula→ ŷ=bx+a

• The goal is to determine values of a and b that best fit the data

• We use those values to predict the values of y given x

Example:

ŷ=20(x)+1000

• Every time that alcohol consumption increases by 1 oz reaction time increases by 20 ms

• The “baseline” reaction times without alcohol are around 1000ms (a=1000)

• Then given a dosage level we can predict ones reaction time on the task

Regression and Error:

• Predictions will always fall on the regression line→ most data will lie beside

• The difference between a data point and the corresponding prediction is known as the prediction error/residual 

Linear Regression coefficients:

Formula:

ŷ=bx+a→ min Σ(y-ŷ)²

Hypothesis Testing:

Is a regression significant?

• Is x a good predictor of y?

• Is b significantly different from 0?

How do we test?

• A regression is significant if the corresponding correlation is significant

Degrees of Freedom:

• We use the two sample means to calculate everything else

• Use N-2 

• Those free data points are reported so that it is clear how many observations we have used to draw conclusions

• DF will always be less than n 

• Used to determine p values and critical values 

Standard Error estimate:

Formula→ S   = S   (1-r²) (N-1) 

        Y-ŷ     y              ------------

                (N-2)

• The proportion of variance in Y we can account for with X equals r²

Constraints:

1. Regression is sensitive to outliers

2. Only appropriate for linear relationships

3. Regression is sensitive to restriction of range

4. Beware of heterogeneous samples

5. Regression does not allow for us to make predictions about causation

Correlation:

Correlation→ degree to which two variables are related

• Positive→ as X increases, Y also increases

• Negative→ as X increases, Y decreases

Linear Relationships→ represented by a straight lines

• Shows there is a common relationship between both variables

Covariance→ degree to which two variables vary together

Formula→ (x-x)(y-Y)

      —-------------------

N-1

Correlation Coefficient→ a statistic that measures the relationship between two variables

• Range from -1 to -1

• Shows type of relationship and its strength

Pearson's product moment correlation→ r

Spearman's rank order correlation→ rs

Point-biserial correlation→ rpb