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Mechanics of Linear Regression Analysis

Build the Linear equation: Use least squares method, regression equation will be a straight line that best fits the data from independent and dependent variables

Statistical Testing: Hypotheses: Ho: B = 0 or Ha: B =/= 0. Two options to calculate test statistic (see notes)

Decision rules: Can use F or t test. F.05(1), dfR, dfr. Tcrit=t.05(2), ny-2

Calculate the Coefficient of Determination: Measurement of how much of the variation in Y is explained in the regression model (line), expressed as a percent. r2 = b(SCP) / SSy

Linear Regression and Correlation

Linear Regression:

Tests for a significant relationship between two variables

Defines each variable: y = dependent variable, x = independent variable. Y varies in response to changes in X.

Defines functional mathematical relationship: y = bx + a

a= y-int, b= slope, can be pos, neg or zero

Used for prediction

Correlation:

Tests for a significant relationship between two variables

Measures the degree of association between the 2 variables (strength of relationship)

Multiple Contrasts Tests

Can be used like a Generic or Control Group Test

But most often, are used to evaluate homogenous groups of means against other such groups or individual means

IE: compare means 1 and 2 to 3, or 1, 2, and 3 to 4, etc.

Most commonly used: Scheffe's test

Scheffe:

Usually comparing two means against one

Statistical hypotheses have a specific format: ie, ho: u3 = (u1 + u2)/2. ha: u3 =/= (u1 + u2)/2

(see notes for specific equations for equal and unequal sample sizes)

Scrit = sq. root of ((k-1) F .05(1), dfg, dfw)

Control Group Tests

Evaluates differences between experimental groups vs. a control group (not differences among experimental groups)

Most commonly used: Dunnett’s Test

Dunnett's:

Can be one or two-tailed

(see notes for specific equations for equal and unequal sample sizes)

Test statistic is q'

q' crit = q'.05(1), dfw, k

Generic Multiple Comparisons Tests

Evaluate all possible pairs/combinations of means

Two most commonly used: Tukey's HSD test, Student-Newman-Keuls (SNK) test

Tukey's:

(see notes for specific equations for equal and unequal sample sizes)

qcrit = q.05, dfg, dfw

SNK:

(see notes for specific equations for equal and unequal sample sizes)

qcrit = q.05, dfw, p

p is range of means being compared

Intro to Multiple Comparisons Tests

Only used when a significant difference is found with ANOVA

Also referred to as “ad hoc” or “a posteriori” tests

Three main types: 1) Generic Multiple Comparisons Tests 2) Control Group Test 3) Multiple Contrasts Tests

Mechanics:

Test Statistic: qobs= difference in numerator/ standard error term in denominator

Statistical Hypotheses: Ho: uA = uB, Ha: uA =/= uB. Arrange all the means in order from largest to smallest, and calculate observed test statistic for each. Because of the enclosure rule (which states that if two means are not different from each other, then all the means in between them are not either)

Reporting results: Arrange means in order and underline the means or subsets of means that are NOT significantly different from each other

ANOVA Mechanics

Statistical Hypotheses: Ho = u1 = u2 = u3 = u4 ... uk

Formulae: Numerator ("among group variance") is the variation among grand mean and the sample means Demoninator ("within group variance") is the variation within each sample, around each sample mean

Critical Values: Fcrit = F.05 (1), dfg, dfw. dfg = degrees of freedom of numerator. dfw = degrees of freedom of denominator.

Results: Fobs > Fcrit, then do not reject. Fobs < Fcrit, then reject null

Next step: If there is no difference, you are done. If there is a difference, you must follow with a mult. comparisons test to determine which means are different.

ANOVA Background

Developed by Fisher

Evaluates differences between three or more means because the t test cannot be applied

Types of ANOVA

One Way ANOVA: Examines one factor at a time, tests for differences among levels of the factor

Fixed effects (Model I): Levels of the factor are specifically chosen (ie pig feed types)

Random effects (Model II): Levels of the factor are randomly chosen

Two Dependent Sample T test (Paired)

Used to compared two dependent population means

Can be helpful in pairing individuals with similar characteristics if you can’t make the same individual undergo two treatments (Matched Pair Design)

Remember, in two sample tests, the variances must be equal

Two Independent Sample T Test (Unpaired)

Used to compared two independent population means

Remember, in two sample tests, the variances must be equal

Student's (Gossett's) T test

Gossett couldn’t use z scores because populations were too small

He created a new test under a pseudonym

New test was leptokurtic

Three main types (One sample, Two sample independent, and Two Sample dependent)

Within these, there are one-tailed and two-tailed variations

One-tailed T test: You want to know the direction of difference (i.e. does a new fertilizer make plants grow taller?) (Note: when it is one tailed, don’t need to divide the .05 into two sections on either side of normal curve; you put the “whole” thing on one end)

Two-tailed T test: Only evaluates whether a difference exists, not the direction (Note: .05 is split up into two rejection regions on either side of normal curve)

Types of Errors

Type I: rejecting a null hypothesis when it is actually true

Type II: not rejecting a null hypothesis when it is false

We can use our p values to express how much confidence we have in our answer.

Additionally, we can report variability around the mean when reporting our answers in a variety of ways.

Mean +/- SD

Mean +/- SE

Mean +/- 95% confidence intervals [Confidence Intervals are zcrit (1.96 when doing 95%) x SE]

One Sample T test

Used to compare a population mean inferred from a sample with a hypothetical population mean.

Using Z-Scores

Z-scores can be used to interpret statistical results.

Rules:

Zobs < Zcrit, then do not reject

Zobs >= Zcrit, then reject

Example: Z critical = 1.96, Z observed = 1.79

In this example, we did not reject null hypothesis. CO concentration of 10.43 is not different from 10 (p>0.05, n=18, z=1.79).

Intro to Z-Scores

Can be used to find the probability of a certain value in a population

Convert data into z-scores

Zi=(Xi-mean)/SE

Use z-score chart on pg 676 of Zar

Example: Probability (P) of X>66 = Probability P of Z>6

(Note: Probability of a score less than a given numberis found by doing1-%)

Normal Distribution and Probability

Bell curves: most values are found near the middle, very few at either extreme

Deviations from Normal Distribution include Asymmetric and Kurtic

Positive: Skewed to right

Negative: Skewed to left

Platykurtic: Flat, more values closer to the mean

Leptokurtic: Peaked, most values towards tails

Probability:

Area underneath the normal curve can be used to find probability of finding values of a variable greater than x in a population, total area under the curve is 1 or 100%

Exploratory Data Analysis (EDA) [Part II]

Measures of variation or dispersion:

Range: Numerical difference between max and min value, problem is that it is subject to sampling error and not a good estimate of population range

Variance: Squared units, Population Variation is represented by o^2, Sample Variance is represented by s^2, to find, divide SS by degrees of freedom (n-1)

Standard Deviation (SD): Measure of spread of data around the mean, to find, take square root of variance

Standard Error (SE): Describes dispersion of sample means around their population mean, to find, divide SD by the square root of n

Coefficient of Variation: Used to compare amount of variation among samples with data that differs in magnitude, expressed as a percentage of the mean, to find, (SD/mean) x 100

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