Applied Statistical Analysis EDUC 6050 Week 6 Finding clarity using data
Today 1. Hypothesis Testing with ANOVA • One-Way • Two-Way 2
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 3
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 4
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 5
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 6
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 7
Do you know comparison population mean ! and standard deviation # ? Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 8
ANalysis Of VAriance • It is a whole class of methods • Generally used with experimental designs • Has similar assumptions to t-test • Gives an “omnibus” result Sum of df Mean Square F p Squares Children 26.8 3 8.92 2.38 .087 Residuals 127.5 34 3.75 9
The F statistic 10
Do you know comparison population mean ! and standard deviation # ? There are several types Know comparison population mean !? Yes No How many groups (or No repeated measures) do you Z-Tests Yes have? 2 3+ Do you have repeated measures? T-Tests ANOVA Yes Paired Samples T- Test Do you have repeated Independent No measures? Samples T-Test Repeated Measures Yes No ANOVA Yes Yes Two-Way ANCOVA Do you have two ANOVA independent variables (two different grouping No No One-Way Do you have continuous or categorical variables)? ANOVA covariates to include in the model? 11
ID Outcome Group General 1 8 1 Requirements 2 8 1 3 9 1 1. Need a DV on an 4 7 1 interval/ratio scale, 5 7 2 2. IV defines 2+ 6 9 2 different groups (or time points) 7 5 2 8 5 2 12
ID Time 1 Time 2 General 1 8 7 Requirements 2 8 8 3 9 6 1. Need a DV on an 4 7 6 interval/ratio scale, 5 7 8 2. IV defines 2+ 6 9 5 different groups (or time points) 7 5 3 8 5 3 13
Hypothesis Testing with ANOVA The same 6 step approach! 1. Examine Variables to Assess Statistical Assumptions 2. State the Null and Research Hypotheses (symbolically and verbally) 3. Define Critical Regions 4. Compute the Test Statistic 5. Compute an Effect Size and Describe it 6. Interpreting the results 14
1 Examine Variables to Assess Statistical Assumptions Basic Assumptions 1. Independence of data 2. Appropriate measurement of variables for the analysis 3. Normality of distributions 4. Homogeneity of variance 15
1 Examine Variables to Assess Statistical Assumptions Basic Assumptions 1. Independence of data 2. Appropriate measurement of variables Individuals are independent of for the analysis each other (one person’s scores 3. Normality of distributions does not affect another’s) 4. Homogeneity of variance 16
1 Examine Variables to Assess Statistical Assumptions Basic Assumptions 1. Independence of data 2. Appropriate measurement of variables for the analysis 3. Normality of distributions 4. Homogeneity of variance Here we need interval/ratio DV 17
1 Examine Variables to Assess Statistical Assumptions Basic Assumptions Normality of 1. Independence of data the residuals 2. Appropriate measurement of variables for the analysis 3. Normality of distributions 4. Homogeneity of variance
1 Examine Variables to Assess Statistical Assumptions F df1 df2 P Basic Assumptions 2.86 3 34 .051 1. Independence of data The variances of groups should 2. Appropriate measurement of variables be equal (not strict if each for the analysis group has similar sample sizes) 3. Normality of distributions 4. Homogeneity of variance 19
1 Examine Variables to Assess Statistical Assumptions Examining the Basic Assumptions 1. Independence: random sample 2. Appropriate measurement: know what your variables are 3. Normality: Histograms, Q-Q, skew and kurtosis 4. Homogeneity: Levene’s Test 20
2 State the Null and Research Hypotheses (symbolically and verbally) Hypothesis Symbolic Verbal Difference between Type means created by: Research One of the groups’ True differences At least one 𝜈 is Hypothesis different than the means is different others than the others Null There is no real Random chance All 𝜈 ’s are the same Hypothesis difference between (sampling error) the groups 21
3 Define Critical Regions How much evidence is enough to believe the null is not true? Before analyzing the data, we define the critical regions (generally based on an alpha = .05) 22
3 Define Critical Regions We decide on an alpha level first Then calculate the critical value (based on sample size) 23
3 Define Critical Regions We decide on an alpha level first Then calculate the critical value (based on sample size) Use the table in the book • Base on alpha and 2 specific df’s 𝒆𝒈 𝒐𝒗𝒏 = 𝒉 − 𝟐 where g is number of groups 𝒆𝒈 𝒆𝒇𝒐 = 𝑶 − 𝒉 24
3 Define Critical Regions We decide on an alpha level first Then calculate the critical value (based on sample size) 𝑮 𝒅𝒔𝒋𝒖𝒋𝒅𝒃𝒎 𝟑, 𝟑𝟘 = 𝟒. 𝟒𝟒 So our critical region is defined as: 𝜷 = . 𝟏𝟔 𝑮 𝒅𝒔𝒋𝒖𝒋𝒅𝒃𝒎 𝟑, 𝟑𝟘 = 𝟒. 𝟒𝟒 25
4 Compute the Test Statistic 𝟑 Source of Variance SS df MS F 𝜽 𝒒 𝑇𝑇 FGHIGGJ 𝑁𝑇 FGHIGGJ 𝑇𝑇 FGHIGGJ 𝑜(∑𝑁 B − ∑𝑁 B Between Groups − 1 𝑒𝑔 𝑁𝑇 GMMNM 𝑇𝑇 FGHIGGJ + 𝑇𝑇 MGPQRSTU FGHIGGJ 𝑇𝑇 MGPQRSTU Within Groups ∑𝑇𝑇 GTVW HMGTHXGJH 𝑂 − (Residual) 𝑒𝑔 MGPQRSTU Total 𝑇𝑇 GTVW HMGTHXGJH + 𝑇𝑇 MGPQRSTU 26
4 Compute the Test Statistic “Sum of Squares” – adding up all the squared deviations (essentially like SD) 𝟑 Source of Variance SS df MS F 𝜽 𝒒 𝑇𝑇 FGHIGGJ 𝑁𝑇 FGHIGGJ 𝑇𝑇 FGHIGGJ 𝑜(∑𝑁 B − ∑𝑁 B Between Groups − 1 𝑒𝑔 𝑁𝑇 GMMNM 𝑇𝑇 FGHIGGJ + 𝑇𝑇 MGPQRSTU FGHIGGJ 𝑇𝑇 MGPQRSTU Within Groups ∑𝑇𝑇 GTVW HMGTHXGJH 𝑂 − (Residual) 𝑒𝑔 MGPQRSTU Total 𝑇𝑇 GTVW HMGTHXGJH + 𝑇𝑇 MGPQRSTU Gets split by where the variation is coming from • Between the groups (the differences in the groups) • Within the groups 27
4 Compute the Test Statistic Within group Outcome variation (residual) Within group variation (residual) 28
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