(Individuals With Disabilities Act Transformation Over the Years)D
- 1. (Individuals With Disabilities Act Transformation Over the Years) Discussion Forum Instructions: 1. You must post at least three times each week. 2. Your initial post is due Tuesday of each week and the following two post are due before Sunday. 3. All post must be on separate days of the week. 4. Post must be at least 150 words and cite all of your references even it its the book. Discussion Topic: Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA. What do you feel are some things that can or should be implemented to better assist with students that have disabilities? Tell me about these ideas and how would you integrate them? 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-
- 2. 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
- 3. 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
- 4. One-way ANOVA One-Way ANOVA Hypotheses statements Test statistic = ������� ����� �������� ���ℎ �� ����� �������� 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
- 5. 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
- 6. One-Way ANOVA Example problem One-Way ANOVA Example problem Studentized Range q table One-Way ANOVA Example problem IntroDesire is to evalute the difference in effect between two or more groupsThe groups are classified as levels of a factorwhen there is only 1 factor, this is called a 1 way ANOVAsamples are random and independent of each other so that group means can be comparedANOVA compares means of groups by analyzing the variation among and within groupstotal variation is subdivided into variation due to differences between or within groupsAssumptionsc groups are independently and randomly selectedsample values in each group are from a normal distributiongroups have equal variances (don't need to worry if sample sizes in each group are the same)Null hypothesis is no difference between meansunder the Null hypothesis means of the c groups are assumed to be equalcalculations sum the squared differences between group means and the grand meancalculations also sum the squared differences between individual group data and their group meansnullalternateLittle Between group variancelots of between group variance Test StatisticThe ANOVA test statistic follows an F distributionF statistic is a ratio of between group and within group variance indiciesalpha - is the level of significance - used
- 7. to determine the critical F valuethe 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 TableStandard deviations for each group can be obtained by taking the square root of the variancesF>Fcrit then REJECT the null hypothesisp-value< .05 then REJECT the null hypothesisAt least one of the group means is significantly different from one other (you don't know which however)c-1 degrees of freedom for between groupn-c degrees of freedom for within groupF = MS for between/ MS for withinFcritical=3.6823203437MS = SS/dfWrite UpResults 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 Example1Is there a meaningful difference in group means?You needalpha0.05GroupAGroupBGroup CAnova: Single Factor7111481412SUMMARY101410GroupsCou ntSumAverage Variance121216GroupA5448.84.771013GroupB56112.23.2Grou p C565135ANOVASource of VariationSSdfMSFP-valueF critBetween Groups49.7333333333224.86666666675.78294573640.0174334 863.8852938347Within Groups51.6124.3Total101.333333333314 multiple comparison testThere are many different types of multiple comparison tests (also called post hoc tests)Example: Tukey HSD Multiple comparison testsimultaneous comparison of means between all pairs of groups (there are other tests that compare more than 2 means simultaeously)assumes equal sample sizesSteps1compute the absolute mean differences among all pairs of group meansthere will be c(c-1)/2 pairs of meansuse the average column values in Excel SUmmary table to obtian the group means2compute the HSD value The studentized range (Q) is calculated for a particular alpha value for c and (n-
- 8. c) --- use the tablen is the group sample sizeMSE is found from the ANOVA summary table (MS within groups column)3compare each of the pairs of mean differences against the HSD valueif the absolute difference in the sample means is greater than the HSD value, then the means are significantly different Example2Tukey HSD Multiple ComparisonsWhat should the minimal difference in group means be to indicate significance?You needLevel of Significance (alpha)0.05c (number of groups - k in this picture)3n (sample size)15n- c12MSE (from ANOVA table)4.3Studentized Range (from table)3.67GroupsCountSumAverageVarianceHSD (use formula)1.9649642914GroupA5448.84.7GroupB56112.23.2Com parisonsAbsolute DIfferenceHSDResults: Is HSD smaller?Group C565135GroupA to GroupB3.41.9649642914YessignificantGroupA to GroupC4.21.9649642914YessignificantGroupB to GroupC0.81.9649642914NoANOVASource of VariationSSdfMSFP-valueF critBetween Groups49.7333333333224.86666666675.78294573640.0174334 863.8852938347Within Groups51.6124.3Total101.333333333314 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
- 9. 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
- 10. • 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
- 11. • 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
- 12. 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
- 13. • 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