Fluent Essays

Applied Learning Research Project Phase II (due August 21) Project Part 2 :  Reference all prior notes and readings for the course as support for completing this assignment.  Ensure that ALL prior instructor requested corrections to your approaches to  formatting,   etc ... taken when completing your HW assignments,  are understood and that you have addressed any misconceptions you had prior to submitting your Project Part 2 assignment Part 2: Simple Regression Pre-Analysis Select 1 response and 1 explanatory variable from the Project 1 dataset (list your choices). Please do not choose variable combinations that were assigned in HW previously Perform a Simple Regression assumption check. That is, check all 4 Simple Regression assumptions.  In 4 separate subheadings (Linearity, Normality, Independence, Equal Variances) write up your results.  Include all supporting evidence (tables, charts etc ...). Appropriate formatting and use of complete sentences is expected. If you were justified in running the simple regression (that is all assumptions for running a simple regression analysis were met) please report your regression output tables from Excel . Also, please run the analysis and interpret your output tables relative to the significance of the regression model, the coefficient of the regression explanatory variable, and the regression numerical summaries (Multiple R and R squared).  Write up your significant/ not significant simple regression analysis results as follows: A simple regression was run to predict (response variable) from (Explanatory variable). Results show that the explanatory variables does / does not statistically significantly predict (response variable). F(df regression, df residual) = Significant F, p< , =, or > .05. The explanatory variable does/ does not contributes significantly to the prediction with p < .05. Finally, write your simple regression equation. Be sure to name the explanatory and response variable when writing the equation. If you were NOT  justified in running the simple regression (that is assumptions for running a simple regression analysis were met) , please provide an explanation why not and support your conclusion using statistical details revealed during your investigation. Write up your results as follows: A simple regression was run to predict (response variable) from (Explanatory variable). Results show that the explanatory variables does not statistically significantly predict (response variable). F(df regression, df residual) = Significant F, p< , =, or > .05. The explanatory variable does/ does not contributes significantly to the prediction with p <, =, or > .05. Design of Experiments Experimental Design • Systematic process to layout, design, and investigate problems in research and industry • Starts with a screening process to identify factors (variables) that may have an impact on responses intended to be observed • May consist of a number of iterative experiments to upgrade and revise factors and details understood regarding the factors • Start by choosing factors that may affect results • Factors that do not change during the experiments are referred to as constants (or controlled) • Optimization follows the screen process where the focus is on establishing an approach for predicting the response variable from the factors identified during the screening process • Formulate hypothesis of the relation between factors and predicted outcomes of the experiment • Construct a statistical model that represents experiment results • Testing follows optimization to ensure procedures are sound and to evaluate the robustness relative to fluctuations in factor conditions Experimental Design • The experimental design specifies the number and type of experiments as well as the factors and their combinations used for each experiment, and how many times experiments will be run (replicated) and in what order. • What measurements to take (responses) is also specified • Specify conditions and assumptions under which experiments are run is determined • What levels (values) of each factor should be examined • Resources and materials needed for the experiments are stated • Run a series of simulations , or actual experiments, to refine factor selection and understanding of factor effects Sections to include in Experimental Design write up • Title • Hypothesis (questions to be investigated) • Design type (simple, factorial, partial factorial) • Factors and Levels of the factors • Response variable and how it is measured • Number of replications • Constraints, assumptions, limitations of the experiment Some frequent responses that are measured • Mean • Standard deviation • Variance • Standard error Type of Experimental Designs • Simple • Start with a initial benchmark configuration and then vary one factor level of a factor at a time and then observe performance • The number (n) of experiments to be conducted is • Where ni is the number of levels for the ith factor. K is the number of factors. For example, in an experiment with 2 factors with 3 and 4 levels respectively n= 1 + (3-1) + (4-1) = 1 + 2 + 3 = 6 simple experiments are to be conducted Types of Experimental Designs • Full factorial design • For each successive experiment or simulation, investigate all possible combinations of all factor levels • The number (n) of experiments to be conducted is • For example, in an experiment with 2 factors with 3 and 4 levels respectively n= 3x4 = 12 experiments are to be conducted Type of Experimental Designs • Fractional Factorial Designs • Uses a subset of factors and factor levels in the experiments • Experimental groupings • Conducting a full factorial design investigation may not be warranted or feasible • Number of experiments to conduct is determined by multiplying together the subset of factor levels of the select factors that will be investigated Power of a design • Used to address issues of experimental accuracy • Determine sample size needed to measure the response at a desired level of accuracy Experimental Design common errors • Ignoring causes of experimental error that may impact results • All important factors, that can impact results, have not been identified • When varying several factors, have a lack of clarity to what the actual effect is • Using multiple simple designs as opposed to a factorial design • Not examining the impact of interactions between factors … if factor level effects are being influenced by the presence of other factor levels Intro Desire is to evalute the difference in effect between two or more groups The groups are classified as levels of a factor when there is only 1 factor, this is called a 1 way ANOVA samples are random and independent of each other so that group means can be compared ANOVA compares means of groups by analyzing the variation among and within groups total variation is subdivided into variation due to differences between or within groups Assumptions c groups are independently and randomly selected sample values in each group are from a normal distribution groups have equal variances (dont need to worry if sample sizes in each group are the same) Null hypothesis is no difference between means under the Null hypothesis means of the c groups are assumed to be equal calculations sum the squared differences between group means and the grand mean calculations also sum the squared differences between individual group data and their group means null alternate Little Between group variance lots of between group variance Test Statistic The ANOVA test statistic follows an F distribution F statistic is a ratio of between group and within group variance indicies alpha - is the level of significance - used to determine the critical F value the null hypotheis is rejected if the value of the test statistic is greater than the critical F value Critical F value 1-way ANOVA Summary Table Standard deviations for each group can be obtained by taking the square root of the variances F>Fcrit then REJECT the null hypothesis p-value< .05 then REJECT the null hypothesis At least one of the group means is significantly different from one other (you dont know which however) c-1 degrees of freedom for between group n-c degrees of freedom for within group F = MS for between/ MS for within Fcritical= 3.6823203437 MS = SS/df Write Up Results of the ANOVA test shows that  a statsitically significant difference in group means (specify the groups) has been detected (F(2,15) = 7.15, p = .007) Note: Practically different and statistically significantly different are not necessarily the same Formulas for values in the ANOVA table MSE Example1 Is there a meaningful difference in group means? You need alpha 0.05 GroupA GroupB Group C Anova: Single Factor 7 11 14 8 14 12 SUMMARY 10 14 10 Groups Count Sum Average Variance 12 12 16 GroupA 5 44 8.8 4.7 7 10 13 GroupB 5 61 12.2 3.2 Group C 5 65 13 5 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 49.7333333333 2 24.8666666667 5.7829457364 0.017433486 3.8852938347 Within Groups 51.6 12 4.3 Total 101.3333333333 14 multiple comparison test There are many different types of multiple comparison tests (also called post hoc tests) Example: Tukey HSD Multiple comparison test simultaneous comparison of means between all pairs of groups (there are other tests that compare more than 2 means simultaeously) assumes equal sample sizes Steps 1 compute the absolute mean differences among all pairs of group means there will be c(c-1)/2 pairs of means use the average column values in Excel SUmmary table to obtian the group means 2 compute the HSD value  The studentized range (Q)  is calculated for a particular alpha value for c and (n-c) --- use the table n is the group sample size MSE is found from the ANOVA summary table (MS within groups column) 3 compare each of the pairs of mean differences against the HSD value if the absolute difference in the sample means is greater than the HSD value, then the means are significantly different Example2 Tukey HSD Multiple Comparisons What should the minimal difference in  group means be to indicate significance? You need Level of Significance (alpha) 0.05 c (number of groups - k in this picture) 3 n (sample size) 15 n-c 12 MSE (from ANOVA table) 4.3 Studentized Range (from table) 3.67 Groups Count Sum Average Variance HSD (use formula) 1.9649642914 GroupA 5 44 8.8 4.7 GroupB 5 61 12.2 3.2 Comparisons Absolute DIfference HSD Results: Is HSD smaller? Group C 5 65 13 5 GroupA to GroupB 3.4 1.9649642914 Yes significant GroupA to GroupC 4.2 1.9649642914 Yes significant GroupB to GroupC 0.8 1.9649642914 No ANOVA Source of Variation SS df MS F P-value F crit Between Groups 49.7333333333 2 24.8666666667 5.7829457364 0.017433486 3.8852938347 Within Groups 51.6 12 4.3 Total 101.3333333333 14 ANOVA ANOVA • Analysis of Variance • Statistical method to analyzes variances to determine if the means from more than two populations are the same • compare the between-sample-variation to the within-sample-variation • If the between-sample-variation is sufficiently large compared to the within-sample- variation it is likely that the population means are statistically different • Compares means (group differences) among levels of factors. No assumptions are made regarding how the factors are related • Residual related assumptions are the same as with simple regression • Explanatory variables can be qualitative or quantitative but are categorized for group investigations. These variables are often referred to as factors with levels (category levels) ANOVA Assumptions • Assume populations , from which the response values for the groups are drawn, are normally distributed • Assumes populations have equal variances • Can compare the ratio of smallest and largest sample standard deviations. Between .05 and 2 are typically not considered evidence of a violation assumption • Assumes the response data are independent • For large sample sizes, or for factor level sample sizes that are equal, the ANOVA test is robust to assumption violations of normality and unequal variances ANOVA and Variance Fixed or Random Factors • A factor is fixed if its levels are chosen before the ANOVA investigation begins • Difference in groups are only investigated for the specific pre-selected factors and levels • A factor is random if its levels are choosen randomly from the population before the ANOVA investigation begins Randomization • Assigning subjects to treatment groups or treatments to subjects randomly reduces the chance of bias selecting results ANOVA hypotheses statements One-way ANOVA One-Way ANOVA Hypotheses statements Test statistic = 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Under the null hypothesis both the between and within group variances estimate the variance of the random error so the ratio is assumed to be close to 1. Null Hypothesis Alternate Hypothesis One-Way ANOVA One-Way ANOVA One-Way ANOVA Excel Output Treatment groups Total 0.391495833 23 p-value is found using the F-statistic and the F- distribution. The p-value for this test is less than 0.05. Reject the null hypothesis and conclude that at least one treatment group mean is statistically different. Multiple Comparison Tests If the ANOVA test of the null hypothesis is rejected, then conclude that not all the means are equal but doesn’t suggest which means are statistically different Multiple Comparison Test One-Way ANOVA Example problem One-Way ANOVA Example problem One-Way ANOVA Example problem Studentized Range q table One-Way ANOVA Example problem