Measure of Central Tendency

Measures of Central Tendency

• Arithmetic Mean or Average
• Median- Middle datum of a sample
• 50% of data lies about mean
• 50% of data lies below mean
• To find the mean
• Step 1- sort data
• Step 2- determine whether n= even or odd
• Mode- datum that occurs most often in a sample
• 2 Step process
• Sort the data
• Conduct frequency analysis
• Count the number of occurrences of each datum

Measure of Variation

Measure of Variation
Used to asses the variation of data around the mean or median or mode

• Range
• the numerical difference between the minimum and maximum values in a data set
• Variance
• calculate in squared unites of the original data
• population
• sample
• Standard deviation
• measure of the spread of data around a mean
• Standard error
• describes dispersion of sample mans around their population mean
• Coefficient of variation
• used to compare amount of variation among samples with data that differs in magnitude

Variance 

• 2 step process
• Find SS
• Divide SS by n-1=df
• use n-1=df 
• Conservative estimate of the population variance 

Normal Distribution

Normal Distribution

• Used to determine the probability of obtaining random samples with different means
• Many samples and populations contain data that fit a normal distribution
• Can be used for basis of statistical testing

Characteristics

• Bell Curve
• most values near the middle datum or average of the sample
• very few values near the upper and lower extremes
• Data fit the formula of a normal distribution
• Y= frequency of a value of x 

Deviations from a Normal Distribution

• Asymmetric Deviations
• Skewed distributions 
• Skewed to the right= pos. skewed
• Skewed to the left= neg. skewed
• Mathematical Analysis of skewness
• Kurtic Deviations 
• Platykurtic
• Leptokurtic
• Mathematical analysis of kurtosis

Student's t-test

Student's t-test

• Background
• developed by Gusset working at Guinness Brewery
• Problems with the Normal Distribution
• Gossett discovers the t distribution
• Publishes it under an assumed name-student

Calculation of Z scores require knowledge of population parameter

• Population Mean
• Population Standard Deviation and Standard Error
• Small samples do not provide reliable enough estimates of population parameters

Characteristics of the t distribution 

• Leptokurtic
• As n and v (df=v=n-1) increased the t distribution begins to approach a normal distribution

Types of Student's t tests

• One-sample Student's t test
• Two independent (unpaired) Samples Student's t test
• Two dependent (paired) Samples Student't test

One-sample Student's t test

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

Two Independent (unpaired) Sample Student's t test

• Used to compare two independent population mean inferred from two samples (independent indicated that the value from both samples are numerical independent of each- there is no correlation 

Two dependent (paired) Samples Student's t test 

• Used to compare two dependent populations inferred from two samples (dependent indicates that the value from both samples are numerically dependent upon each other- there is a correlation between corresponding values)

Two variations of all Student's t test

• Two-tailed test
• One-tailed test 

Two-tailed test- evaluates whether a difference exists between 2 samples, not the direction of the difference

One-tailed test- evaluates whether a difference exists between 2 samples, and specifically evaluates the direction of the difference 

ANOVA

ANOVA

• Developed by Fisher
• Studied Agriculture and crop output with different fertilizers 
• Needed test that could evaluate differences between three or more means
• Why- problems with applying the Student's t test

One- ANOVA

• Examines one factor at a time, tests for differences among levels of the factor

Fixed Effects or Model I One-way ANOVA

• the levels of the factor are specifically chosen by the investigator

Random effects or Model II One-way ANOVA

• the levels of the factor are randomly chosen by the investigator

Mechanics of One-way ANOVA

• Statistical hypothesis
• Formulae
• Critical values and decisions to reject/ not reject the null hypothesis

Formulae

• Focus in on analysis of variance- comparison between 2 types of variance
• Numerator= among group variance (variation among the grand mean and the sample means)
• Denominator= within group variance (sum of the variation within each sample-around each sample mean)

Results of ANOVA

• No difference between among group variance and within group variance
• There is no difference among means
• Stop all testing and write the results section
• Differences exists between among group variance and within group variance
• There is a difference among means
• Follow with a multiple comparisons test to determine which means are different from each other

Example of an ANOVA

1. State the biological question
2. Translate into statistical hypotheses
3. State the alpha level
4. State the statistical test
5. State the assumption of the test
6. Calculate the observed test statistic
7. Find the degrees of freedom and critical value
8. Compared the observed and critical value 
9. Interpret the results 

Multiple Comparisons Test

Multiple Comparisons Tests

• Only used if the results of an ANOVA yield significant difference
• ANOVA results only indicate that a difference exits among means, not where the difference is
• Referred to as ad hoc or a posteriori test 
• Used after you know there is a significant difference from the ANOVA
• Several types of multiple comparisons tests
• Three broad categories
• Generic multiple comparisons test
• Control group test
• Multiple contrasts tests
• Generic Multiple comparisons test
• evaluate all possible pairs/combinations of means
• Tukey's HSD test
• Student-Newman-Keels (SNK) test
• Control Group test
• Evaluate differences between experimental group versus the control group
• Dunnett's test
• Multiple Contrats tests
• can be used like the traditional tests mentioned above to evaluate differences among pairs of mean but is better used to evaluate homogeneous groups of means against other such groups or individual means 
• Scheme Test
• Basic mechanics of Multiple Comparisons Test
• Observed Test Statistic
• SE=standard error
• Statistical Hypotheses
• A and B represent any pairs of means
• Pairwise comparisons 
• arranged in order from largest to smallest
• Calculate observed test statistic for each comparison
• Enclosure Rule
• If two mean are not different from each other then all means in between them are also not different from each other
• Similarities among different Multiple Comparisons Test
• All test involve pairwise comparisons of means
• Rank order the means for comparisons 
• Calculate an observed q value similar to the t test and z scores
• Compare with a critical value and reject or do not reject the null hypothesis for each pairwise comparison 
• Use the enclosure rule in all tests
• Differences among different Multiple Comparisons test
• how the means are rank order
• most test are two-tailed but control group tests can be one tailed test
• the SE term differs among tests 
• Mechanics 
• Arrange statistical hypotheses 
• Calculate test statistics= observed q
• Decisions rules and critical values
• Test 
• Tukey's HSD
• SNK
• Dunnett
• Scheme

Linear Regression

Linear Regression

• Tests for significant relationship between 2 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
• Used for prediction 
• Relationship is a functional dependence
• The magnitude of a dependent variable (Y) is dependent on magnitude of an independent variable (X)
• Functional dependence is a mathematical relationship that can be quantified 
• Linear Regression equation used to describe the mathematical relationship 
• Y=a+bX or 
• Y=bx+a
• Y or dependent or criterion or response variable
• X independent or predictor or regressor variable 
• a= y-intercept (where x=0)
• b= slope or regression coefficient 
• Functional dependence is a mathematical relationship that can be quantified
• Postitive, negative, and no relationship