Experimental Designs Questions - Assignment Example

Experimental designs s: Question 2. A randomized ANOVA is a design in which the s are randomly assigned such that each of the subjects receives precisely one p treatment. This kind of ANOVA has three sources of variation namely treatment, error and total. The treatment source measures the variations between the results of the treatment i.e. the explained variations. The error source collectively measures the variations in scores within the respective treatments i.e. the unexplained variations. The total source measures the variations of the n scores with no consideration paid to the location of the scores within the treatments. Generally, a randomized ANOVA that is complete entails an independent random selection of variables/observations for each of the levels for one factor. On the other hand, a repeated-measures ANOVA compares the means of three or more matched groups with the aim of controlling experimental variability i.e. it tests the equality of means. The matching of the groups should form part of the experimental designs and not something that is done after collecting data. The results that are obtained from repeated-measures ANOVA can only make sense when the subjects of the test are independent. Consequently, repeated-measures ANOVA assumes that each measurement is the sum of an overall mean, a treatment effect, an individual effect and a random component. However, the repeated-measures ANOVA tests are only done when all the random sample members are measured under different conditions (Rutherford, 2011). In conclusion, a randomized one-way ANOVA is used for between-participants designs while a repeated-measures ANOVA is used for correlated-groups designs. The term one-way with respect to ANOVA refers to the procedure in which the ANOVA factor that describes the cause of variations in data is only one. A one-way ANOVA simply compares three or more groups that are defined by one factor. Basically, this one-way ANOVA determines whether there are any significant differences between the means of three or more independent (unrelated) means or groups at a single moment in time. The one-way ANOVA compares the means between these groups and determines whether one of them is significantly different from the others (Rutherford, 2011). The assumptions that are put into contemplation when carrying out this one-way ANOVA are that: i. The population from which samples are obtained is normally or approximately normally distributed. ii. Samples must be independent. iii. The population variances must be equal. 4. In statistics the problem of multiple comparisons occurs when a person considers a set of statistical inferences simultaneously or infers the selected parameters subsets based on the observed values. However, in multiple comparisons, the use of various procedures for protection against Type 1 errors can be applied. An example of such is the Bonferroni method. The error rate in any experiment is the expected number of type 1 errors in a particular group of statistical significance tests. It is computed as Error rate = c (α) c represents the number of comparisons; α, represents the significance levels For instance, in an experiment with 20 independent statistical comparisons at p = 0.05 confidence level, EP = 1. With the Bonferroni adjustment, this implies that, at 0.05 confidence level, type 1 error is expected in the 20 tests for statistical significance. In an experiment-wise error rate, at 0.05 significance levels for the 20 tests, the error rate becomes 1 – (1 – 0.05)20 = 0.64. This indicates that the probability of at least one type 1 error that occurs among these tests is 0.64. Therefore, the suggested alpha level becomes 0.64. 6. Post hoc comparisons are performed generally after obtaining the significant omnibus F. This is after looking at all the probable pair-wise or all feasible pair-wise and otherwise comparison with a main focus on the largest difference between the levels. Usually, the post hoc comparisons should be done after the experiment has been concluded for the patterns that were not specified in priori. The post hoc analyses are mostly concerned with finding the patterns and relationships between subgroups of populations that are sampled and which would have otherwise remained undetected and undiscovered, were reliance to be strictly put upon a priori statistical method (Quinn & Keough, 2002). 8. Repeated-measures ANOVA is more powerful than the randomized ANOVA due to the fact that the primary strength of a repeated-measure design makes an experiment more efficient. Consequently, repeated-measures ANOVA helps in keeping the variability low. The effect of is that it helps in keeping the legality of the results much higher but, still allows for smaller than usual subject groups. 10. a. Source df SS MS F-value Between groups 1 22.167 22.167 3.0057 Within groups 2 14.750 7.375 Total 3 36.917 n = 4; k = 1 Mean square (MS) = Sum of squares (SS) ÷ Degree of freedom F-value = MSBetween groups ÷ MSWithin groups As this is a case of just one predictor variable, degrees of freedom are as shown above in red highlights. Question 2 An F-ratio refers to a ratio that is used to establish whether the variances in two samples that are different are equal. When the F-ratio is not statistically significant, then one may assume that homogeneity of the variances exist and as such employ a standard t-test for the difference of the means. On the other hand, when the F-ratio is found to be statistically significant, then an alternative statistical t-test computation method such as the Cochran and Cox method can be used. Basically, an F-ratio is associated with the variances of two independent samples. Technical terms in the answer 1. Homogeneity of a variance: Refers to a condition whereby all the variables in a sequence have limited or the same finite variance. When homogeneity is determined to hold true for a statistical model another simpler statistical approach called homoscedasticity may be used due to low uncertainty levels. An assumption of the homogeneity of the variance is that variances within a population are equal i.e. the assumption of the analysis of variance (ANOVA) (Jackson, 2011). 2. Statistical significance: This refers to whether any of the differences that have been observed between two variables are real or are just by chance. The significance is calculated as a probability that an observed effect in a research is merely by chance i.e. the tests describe the probability that what is thought of in a relationship between two variables is but a chance occurrence. The test for significance tells about the probability of making an error by assuming that a relationship exists between two variables. 3. Cochran and Cox method: This is a method for the approximation of the probability level of a t statistic as the worth of p such that t = [(w1t1+w2t2) / (w1+w2)] whereby, t1 and t2 are the significant values of the t distribution equivalent to a significance level of p and model sizes of n1 and n2, respectively. The amount of degrees of freedom is indeterminate when n1 ≠ n2. Question 3 Error variance refers to the other sources of variability, other than the systematic variance, in the collection of data that are lumped into one indistinct mass. Error variance has got little to do with “error” although, at times, the variability due to errors can form part of the error variance. Basically put, an error variance refers to all the other sources of variability upon which one is not focusing their attention on (Hedeker, 2006). When the error variance and systematic variance are combined, what is obtained is the total variance. Total variance = Systematic variance + Error variance In experimental designs, a good experiment is that which has got no confounding but, a small error variance, which is relative to the treatment variance. Usually, the main interest is in the qualification of the proportion or part of total variability that is due to the factors being investigated (Kirk, 2013). This proportion of variance can be accounted for as a percentage by a simple division calculation as below: Systematic variance x 100% Total variance Calculating the error variance can be obtained easily by subtracting systematic variance from the total variance. Since error variance measures all variability sources that are not due to the factor under investigation, determining its value will simply entail measuring the variability under investigation within each group. This may likely provide two estimates of the error variance. In deriving one estimate from the two, one should apply the weighted average concept. The average of the two estimates is weighted so that if when one estimate is superior than the other, then its error variance is given much weight (Mukhopadhyay, 2009). Question 4 An independent variable is a variable whose variation does not depend on that of another variable. Scientifically, it is the variable changed or manipulated while testing or observing for an effect on the dependent variable (Quinn & Keough, 2002). A researcher would want to have more than two levels of an independent variable as this allows for the study of the interrelationships between the two independent variables. However, this calls for the researcher to be more careful while designing these experiments. Basically, having two independent variables and then analyzing them one by one would be more ineffective from a scientific point of view and appear incredibly wasteful. Consequently, having two independent variables allows for the simultaneous optimization of the many different responses. Additionally, having more than two independent variables allows for the elimination of the problem of differential influence of any kind and all extraneous variables (Ott & Longnecker, 2010). Question 5 In the analysis of variance (ANOVA), the independent variable can consist of any number of groups or levels. When faced with a definite independent variable and a constant dependent variable and more than two groups of the independent variable (which is a case that would require a multi-way ANOVA and not a one-way ANOVA), then the most appropriate analysis would be that which entails undergoing two steps. These are carrying out of the F-test so as to determine whether there are any significant differences among the means. If the F-score is statistically significant, then a second step is carried out which compares the two means. In this case, a higher within groups variance in ANOVA would imply that there is no significant F-test. The implication of this would mean that the variable in question had no real effect. The within group variance is the error term (SSwithin) while that for between groups is SSbetween. The ‘Within groups’ variance is the variance that is within groups, and it is not due to the independent variables. While analyzing this, the ANOVA and the F-Score yields a ratio of explained variance versus the error variance (Quinn & Keough, 2002). Question 6 In the analysis of variance (ANOVA), once one has obtained a significant F-value, the work is not yet over as the significant F-value only shows that the null hypothesis should be rejected i.e. the means of the test are not equal. At this level, it is still hard to determine the means that are significantly different from the others. A careful examination of the numbers must, therefore, be done so as to determine where exactly the significant differences are. In this context, a more appropriate method would be to conduct a post-hoc test (Härdle & Simar, 2011). There are various post-hoc tests that can be used but, the most common among them is the Tukey’s Honestly Significant Difference (HSD) test. This is because it is easily calculated, versatile and allows for the answering of just about any ANOVA question one may have. Consequently, post-hoc tests are used when the omnibus ANOVA finds a significant effect. The post-hoc tests allows for the determining of the F-value supposedly when it turns to be non-significant. The post-hoc tests also help to keep the ANOVA experiment wise error rates to acceptable levels. Post-hoc tests are also useful in ANOVA since it is more stringent than the regular t-tests. This is due to the fact that the more tests one carries out, the more probable it is that a significant difference will be found by chance. Question 7 Probabilistic equivalence does not imply that two groups are equal to each other but that, the odds of finding a pretest difference between the two groups are perfectly known. Probabilistic equivalence does not imply that there shall be the same mean for the two groups but, that the odds for the two means not being equal being known. Probabilistic equivalence is achieved through the mechanism of random assignment to the groups. When the groups are randomly assigned to, it becomes possible to calculate the probability that the two groups will differ because of the chance alone i.e. because of the random assignment. Probabilistic equivalence is important since after knowing the odds perfectly by observing the chance difference, a person can control it through analysis of the variables through alpha calculation. By randomly assigning the two groups, it is possible to assume that the two groups have a form of equivalence (Härdle & Simar, 2011). They might be equal but, might be probabilistically equal. References Härdle, W., & Simar, L. (2011). Applied multivariate statistical analysis. Berlin: Springer. Hedeker, D. (2006). Longitudinal Data Analysis. Hoboken: John Wiley & Sons. Jackson, S. L. (2011). Research methods: A modular approach. Belmont, CA: Wadsworth/Cengage Learning. Kirk, R. E. (2013). Experimental design: Procedures for the behavioral sciences. Thousand Oaks: Sage Publications. Mukhopadhyay, P. (2009). Multivariate statistical analysis. Singapore: World Scientific. Ott, L., & Longnecker, M. (2010). An introduction to statistical methods and data analysis. Belmont, CA: Brooks/Cole Cengage Learning. Quinn, G. P., & Keough, M. J. (2002). Experimental design and data analysis for biologists. New York: Cambridge University Press. Rutherford, A. (2011). ANOVA and ANCOVA: A GLM approach. Hoboken, N.J: Wiley. Read More