STSCI 2150 final

ANOVA

analysis of variance; compares means between three or more groups

F value (ANOVA)

Mean Square of X / Mean Square of Error

Mean Squares

sum of squares divided by degrees of freedom

Degrees of freedom in ANOVA table

group= g-1
error= n-g
total= n-1

ANOVA assumptions and how to check

1. random samples- look at study
2. normal distribution for measurements within groups- histogram or qq plot
3. variability in groups is about the same (10%)- Levene's test

Steps for an ANOVA test

1) Set-up and Assumptions
2) Complete ANOVA table
3) Statistical conclusion (Reject or not reject)
4) Plain English conclusion

Sum of squares between groups

sum of size of group times (group mean-overall mean)

Sum of squares total

the sum of squared differences between each individual score and the grand mean of all scores

F distribution

The distribution that models the ratio of two variance estimates; used in ANOVA for obtaining the P-value for testing equality of three or more means

r code for p value (F)

1- pf(test stat, df1, df2)

r code for F statistic

qf(1-alpha, df1, df2)

r code for p value (slope)

2(1- pt(test stat, df))

r code for test stat (slope)

qt(1- alpha, df)

What to do if null is rejected (ANOVA)?

Figure out which means differ
- Bonferroni Connection
- Tukey Kramer

Bonferroni correction

suggests that a more stringent significance level is more appropriate for these tests: 𝛼∗ = 𝛼/𝐾where 𝐾 is the number of comparisons being considered.
𝐾= g(g-1)/2
After figuring out new alpha, then do multiple two-sample t tests

Planned comparison

A comparison of differences across levels of an independent variable when the researcher decides during the design of the study to make the comparison rather than waiting until after preliminary data analysis.
Use t dist after ANOVA

unplanned comparisons

Comparison between means that is not directed by your hypothesis and is made after finding statistical significance with an overall statistical test (such as ANOVA).
Use Tukey Kramer after ANOVA

Standard error for planned comparisons

The square root of MSE1/n1 + MSE2/n2

Planned comparison 95% confidence interval

Y2-Y1 +/- (t0.05(2), N-K)(SE)

r

Correlation coefficient; describes the linear relationship between two variables

Correlation assumptions

1. randomly sampled
2. bivariate normal distribution
- linear relationship between X and Y
- cloud of points in scatterplot of X and Y has circular or elliptical shape
- frequency distributions of X and Y are normal

Common deviations from a bivariate normal distribution

funnel, outlier, non linear (can see these in scatterplots)

Spearman's rank correlation

measures the strength and direction of the linear association between the ranks of two variables

residual plot

a scatterplot of the regression residuals against the explanatory variable

residuals

the difference between an observed value of the response variable and the value predicted by the regression line

Interpretation of slope

For every 1 change in x, the predicted y is associated with an increase or decrease on average by the slope.

Interpretation of the intercept

when x = 0, y is expected to equal the intercept

Conditions for least-squares line and how to check

1. random sample- look at study
2. linearity between x and y- look at scatterplot- elliptical cloud/residual plot shows no curved pattern
3. normally distributed residuals- qq plot of residuals/histogram of residuals
4. constant variance- scatterplot- no funnel/hourglass shape

Homoscedasticity

A regression in which the variances in y for the values of x are equal or close to equal

R^2

the proportion (percent) of the variation in the values of y that can be accounted for by the least squares regression line

Interpretation of R^2

Approximately r^2% of the variability in y can be explained by x

confounding variable

a factor other than the independent variable that might produce an effect in an experiment

prospective study

identifies individuals and collects information as events unfold

retrospective study

collect data after events have taken place

Ways to eliminate bias in experiments

- Controls
- Random assignment to treatments
- Blinding
- Random sample

Ways to reduce sampling error

replication, balance, blocking

Randomization

a process of randomly assigning subjects to different treatment groups

Blinding

a technique where the subjects do not know whether they are receiving a treatment or a placebo

Replication

Carry out a study on multiple independent objects

Balance

Nearly equal sample sizes in each treatment

Blocking

grouping of experimental units that have similar properties; within each group, different experimental treatments are applied to different units

When to use test of single proportion?

one categorical variable with two categories
equal to a certain proportion/percent

When to use goodness of fit test?

one categorical variable with more than two categories
each category has certain percent or equal proportions

When to use CI for single mean

one numerical variable
how does a value differ from a mean

When to use test of association?

Two categorical variables
Does one categorical variable influence the other?

When to use linear regression?

Two numerical variables
Does one numerical variable influence the other?

When to use Levene's Test?

Comparing variances

When to use paired t-test?

Comparing two means that aren't independent

When to use two-sample t-test

Comparing two means that are independent

When to use ANOVA?

Comparing three or more means