STAT 430 EXAM

Statistical Inference

Provides methods for drawing conclusions about a population from sample data

Two most common types of statistical inference

Confidence Intervals and Tests of Significance

Conditions for Inference about the Mean

1) We must have a simple random sample
2) We must have normally distributed data
3) We must know the population standard deviation sigma

Confidence Interval Estimate

An interval of likely values for the population parameter determined from sample information

Confidence Level

The measure of one's confidence that the true parameter value lies within the interval

Margin of Error

z_alpha/2 * (sigma/SQRT(n))

How to make the confidence interval narrower?

Increase the sample size or decrease the level of confidence

How to interpret a confidence interval?

"We estimate that the __parameter__ is between __lowerbound__ and __upperbound__ __units__ with a p% confidence interval"

Hypothesis Testing

The process of using sample data to test a claim about the value of a population parameter

Null Hypothesis

Statement of equality and statement of no difficulty

Alternative Hypothesis

Statement of inequality, represents the claim that we seek evidence for

Right-Tailed Test

Tests whether the parameter is either equal to vs greater than some value

Left-Tailed Test

Tests whether the paramete is either equal to vs less than some value

Two-Tailed Test

Tests whether the parameter is either equal to vs not equal to some value

Type I Error

We reject the null when the null is true

Type II Error

We fail to reject the null when the null is false

Z-Statistic

z = (xbar - mu)/(sigma/SQRT(n))

When to reject the null?

If p-value < alpha, reject the null hypothesis

When to fail to reject the null?

If p-value> alpha, fail to reject the null hypothesis

What does the p-value measure?

How unlikely the observed sample mean is, probability of committing a Type I error based on your sample

Null Hypothesis of a T-Test

Ho: mu_a = mu_b

Alternative Hypothesis of a T-Test

Ha: mu_a > mu_b
OR
Ha: mu_a < mu_b
OR
Ha: mu_a =! mu_b

Conditions for a T-Test

1) The two groups being compared must be independent
2) The theoretical distribution of the sampling means should be normally distributed
3) The variances of the two groups should be approximately equal

What procedure do we use in SAS to get a T-Test?

PROC TTEST

How can we know if variances are equal?

Check the Equality of Variances section, which runs a hypothesis test with the null hypothesis of the variances being equal

If variances are equal, what method of t-test is used?

Pooled

How can we randomly assign treatments to individuals in SAS?

1) Use a random number generator to assign a value to each of the individuals
2) Use a PROC RANK to split the data into the desired number of groups

What is the format required for PROC RANK?

PROC RANK DATA=NAME GROUPS=n;
OUT=Output_name;
VAR variable_to_rank;
RUN;

Wilcoxon Rank-Sum Test

Nonparametric test
Ho: The two distributions are identical
Ha: Distributioons are not identical (Dist A is SHIFTED from Dist B)

How do we compute a Wilcoxon Rank-Sum test in SAS?

PROC NPAR1WAY

Paired T-Test

Testing differences between two dependent sample means

How to run a paired T-Test in SAS?

PROC MEANS on the difference with options T and PRT
OR
PROC TTEST on the difference

Linear Correlation Coefficient

Provides a descriptive measure of the strength fo relationship between variable x and y, has a range of values between -1 and 1

Direct Correlation

R = 1

No Correlation

R = 0

Inverse Correlation

R = -1

How to manually calculate R?

Sxy / SQRT(Sxx * Syy)

Coefficient of Determination

r^2, can be interpreted as the proportion of variance in one of the variables that can be explained by variation in the other variable

Important assumption about a correlation coefficient of determination

Each pair of x, y data points is independent of any other pair

What procedure do we use to test for correlation?

PROC CORR

What type of correlation is run by default in a PROC CORR?

Pearson Correlation

How to get a partial correlation?

Use a Partial var; statement in a PROC CORR

Why use a partial correlation?

It controls for the effects of a certain variable, i.e. it finds the relationship between two variables when the effect of the other variables have been removed

What is a Spearman Correlation?

A non-parametric correlation coefficient that looks at the correlation between the ranks of each of the observations oof each of the variables instead of individual values

Least-Squares Criterion

The straight line that best fits a set of data points is the one having the smallest possible sum of squared errors

Residuals

The vertical distance form the actual observed point to the expected value on the line

Regression Line

The straight line that best fits a set of data points according to the least squares criterion

Regression Equation

The equation of the regression line

What is the difference between correlation coefficient and regression?

Correlation coefficient is an index while regression is mathematical function

Format of the Regression Equation

y = B0 + B1*x
where y = dependent variable
B0 = intercept
B1 = coefficient of independent variable
x = independent variable

How to find B1_hat manually?

Sxy/Sxx

How to find B0_hat manually?

y - B1_hat*x

Extrapolation

Trying to use a regression line to predict values outside of the range of observed values

Outliers

Outliers are data points that lie far from the regression line and outside the overall pattern of the data, we need to identify and remove them

Influential Observation

A legitimate data point whose removal causes the regression equation to change considerably, we must determine the reason for an influential observation and try to control for it, rather than removing the point

Why is the sum of residuals always 0?

Because we are using the least squares criterion

What are the two components of a valid regression model?

Deterministic portion and stochastic error

Deterministic Portion

The part that is explained by the predictor variables in the model, all of the explanatory/predictive information of the model should be in this portion

Stochastic Error

The randomness of the error between observed values and expected values (residuals)

What do regression residuals estimate?

The true error

What does random error look like in SAS?

1) The residuals should not be either systematically high or low
2) The residuals should be centered on zero throughout the range of fitted values
3) Random errors should produce residuals that are normally distributed

What does it mean if residuals are not random?

The deterministic portion is not capturing all of the possible explanatory information. This could mean a variable is missing, a higher-order term of a variable in the model is missing, or an interconnection between terms already in the model is missing

What procedure is used to run a simple linear regression?

PROC REG

How do we find the values for our regressions equation?

The parameter estimates section

How do we know which parameters are good?

Check the p-value of each parameter estimate, which is the result of the hypothesis test that the parameter is equal to 0

If the residuals look curved, how do we try to fix it?

Add a quadratic term

What are ways we can transform the data?

Use the log of the data

What should we look at when trying to determine if the model is good?

1) R-Square or Adjusted R-Sq being close to 1, Adjusted R-Sq is used when there are multiple variables and it adjusts itself for interactions between the variables
2) Parameter estimates and the p-value of each
3) ANOVA table p-value
4) Residual analysis, check they are normally distributed and random

Multiple Regression Analysis

Relates two or more independent variables to one dependent variable

In multiple regression, what types of variables must we have?

Dependent variable must be continuous, and independent variables can be continuous or categorical

What are the two types of multiple regression?

Nonexperimental regression which deals with samples of subjects that have a variety of naturally occurring variables
Designed regressed controls the levels of the of the independent variable experimentally

Assumptions for Simple and Multiple Regression

1) The dependent variable must be continuous
2) The data must be independent and identically distributed
3) The errors must be normally distributed with a mean of zero and a standard deviation of sigma squared

How to turn a categorical variable into numeric values?

Use n-1 dummy variables, one will have all 0's, the others will have a 1 at a unique index and 0's everywhere else

What is another procedure that you can use to create a linear regression model?

PROC GLM

How to see different models of multiple regression?

Use the SELECTION = option

How do we select the best regression model by adjusted R-Sq value?

SELECTION = ADJSQ

How to select the best regression model by adding in the variable with the next largest F value as long as the p-value is less than .5?

SELECTION = FORWARD

How to select the best regression model by adding in variables with the next largest F statistic, but if the p-value gets too high (above .15) it removes it?

SELECTION = STEPWISE

How to select the best regression model by choosing the best model at each level of variables with the best R-Sq value?

SELECTION = MAXR

Cooks D Statistic

A statistic that detects outlying observations by evaluating all the variables simultaneous, a Cooks D value greater than the absolute value of 2 should be investigated

One-Way ANOVA

An inferential method used to test the equality of three or more population means

The Null Hypothesis of a One-Way ANOVA

Ho: the population means of each group is the same

The Alternative Hypothesis of a One-Way ANOVA

Ha: at least one population mean is different from the others

Why don't we run a bunch of t-tests on the population means?

1) We have nC2 tests to run which gets big very quickly
2) The probability of making a type I error increases significantly with each test

Requirements of a One-Way ANOVA

1) There must be k simple random samples
2) The k samples are independent of each other
3) The sampling distributions of the sample mean are normally distributed
4) The populations must have the same variance

Between-Sample Variability

The variability among the sample means

Within-Sample Variability

The variability of each sample

How do between-sample variability and within-sample variability relate to ANOVA?

If the between-sample variability is large relative to the within-sample variability, we have evidence to suggest that the samples come from populations with different means

What procedure is ran to get the results of an ANOVA?

PROC ANOVA

What is a post hoc test?

Additional comparisons between the means to determine which means differ significantly

Requirements for Tukey's Test

1) There are k simple random samples from k populations
2) The k samples are independent of each other
3) The populations are normally distributed
4) The populations have the same variance

How do we run a Tukey's Test?

Use the TUKEY option in the means line

Factorial Design

We have two factors that have multiple different levels within each

Crossed Factors

If all levels of factor A combine with all levels of factor B, the factors are crossed

Main Effects

The main effect of factor A is the change in the dependent variable that results from change in level of A
The main effect of factor B is the change in the dependent variable that results from change in level of B

Interaction Effect

There is an interaction effect if the effect of factor A on the dependent variable varies with factor B

What hypotheses are tested in two-way ANOVA?

INTERACTION EFFECT:
Ho: There is no interaction effect between the factors
Ha: There is an interaction effect between the factors

MAIN EFFECTS:
Ho: There is no effect of factor A on the response variable
Ha: There is an effect of factor A on the response variable

Ho: There is no effect of factor B on the response variable
Ha: There is an effect of factor B on the response variable

What happens if the null hypothesis regarding the interaction effect is rejected?

We do not interpret the result of the hypotheses involving main effects because this interaction clouds the interpretation of the main effects

Requirements for Two-Way ANOVA

1) The populations from which the samples are drawn must be normal
2) The samples are independent
3) The populations all have the same variance

How to run a two-way ANOVA in SAS?

In the MODEL statement of PROC ANOVA, use DV = IV1*IV2