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However buy doxycycline 200mg cheap antibiotic guidelines, two values of df determine the shape of an F-distribution: the df used when computing the mean square between groups 1dfbn2 and the df used when computing the mean square within groups 1dfwn2 purchase doxycycline 200 mg antibiotic resistance evolves in bacteria when quizlet. There- fore discount doxycycline 200mg without prescription bacteria icd 9 code, to obtain Fcrit buy doxycycline 100 mg mastercard antibiotics for uti delay period, turn to Table 5 in Appendix C, entitled “Critical Values of F. For df above 50, compare Fobt to the two critical values for the df in the table that bracket your df, using the same strategy we discussed for t-tests in Chapter 11. Our H0 says that Fobt is greater than 1 because of sampling error and that actu- ally we are poorly representing no relationship in the population. However, our Fobt is beyond Fcrit, so we reject H0: Our Fobt is so unlikely to occur if our samples were rep- resenting no difference in the population that we reject that this is what they represent. Therefore, we conclude that the Fobt is significant and that the factor of perceived diffi- culty produces a significant difference in mean performance scores. Of course, had Fobt been less than Fcrit, then the corresponding differences between our means would not be too unlikely to occur when H0 is true, so we would not reject H0. Then, as usual, we’d draw no conclusion about our independent variable, one way or the other. Therefore, the means of the conditions crit ©X 5 17 ©X 5 31 ©Xtot 5 48 differ significantly. However, we do not know whether every increase in difficulty produces a significant drop in Performing Post Hoc Comparisons 307 performance. Therefore, we must determine which specific means differ significantly, and to do that, we perform post hoc comparisons. Fisher’s Protected t-Test Perform Fisher’s protected t-test when the ns in all levels are not equal. We are testing H0: 1 2 2 5 0, where X1 and X2 are the means for any two levels of the factor and n1 and n2 are the corresponding ns in those levels. For example, we can compare the mean from our easy level (8) to the mean from our diffi- cult level (3). Filling in the formula gives 8 2 3 tobt 5 1 1 7 a 1 b B 5 5 Then 15 15 15 tobt 5 5 5 512. To complete these comparisons, perform the protected t-test on all possible pairs of means in the factor. Thus, we would also test the means from easy and medium, and the means from medium and difficult. Find the value of qk in Table 6 in Appendix C, entitled “Values of Studentized Range Statistic. Ignore whether differences are positive or negative (for each pair, this is a two-tailed test of H0: 1 2 2 5 0). The means from the easy level (8) and the difficult level (3) differ by more than 4. The mean from the medium level (6), however, differs from the other means by less than 4. If these two conditions were given to the population, we would expect to find one population for easy with a around 8 and another population for difficult with a around 3. We cannot say anything about the medium level, however, because it did not produce a significant difference. Finally, as usual, we would now interpret the results in terms of the behaviors being studied, explaining why this manipulation worked as it did. If Fobt is larger than Fcrit, then Fobt is significant, indicating that the means in at least two conditions differ significantly. If Fobt is significant and there are more than two levels of the factor, determine which levels differ significantly by performing post hoc comparisons. If you followed all of that, then congratulations, you’re getting good at this stuff. The Confidence Interval for Each Population As usual, we can compute a confidence interval for the represented by the mean of any condition. This is the same confidence interval for that was discussed in Chapter 11, but the formula is slightly different. Follow the same procedure to describe the from any other significant level of the factor. Note that we include the medium level of difficulty, even though it did not pro- 10 duce significant differences. The way to 0 Easy Medium Difficult do this is to compute the proportion of variance Perceived difficulty accounted for, which tells us the proportional improve- ment in predicting participants’ scores that we achieve by predicting the mean of their condition. Thus, 2 reflects the proportion of all differences in scores that are associ- ated with the different conditions. The larger pb the 2, the more consistently the factor “caused” participants to have a particular score in a particular condition, and thus the more scientifically important the factor is for explaining and predicting differences in the underlying behavior. Because 43% is a very substantial amount, this factor is important in determining participants’ performance, so it is important for scientific study. Recall that our other measure of effect size is Cohen’s d, which describes the magni- tude of the differences between our means. However, now we are getting to more complicated designs, so there is an order and logic to the report. Typically, we report the means and standard devia- tion from each condition first. A significant Fobt indicates that the means are unlikely to represent one population mean. Then we determine which sample means actually differ significantly and describe the relationship they form. All of the research designs in this book involve one dependent variable, and the statistics we perform are called univariate statistics. We can, however, measure participants on two or more dependent variables in one experiment. Even though these are very complex procedures, the basic logic still holds: The larger the tobt or Fobt, the less likely it is that the samples represent no relationship in the population. The program also computes, the X, s , and 95% confidence interval for for each level. A one-way analysis of variance tests for significant differences between the means from two or more levels of a factor. The experiment-wise error rate is the probability that a Type I error will occur in an experiment. Fobt is computed using the F-ratio, which equals the mean square between groups divided by the mean square within groups. Fobt may be greater than 1 because either (a) there is no treatment effect, but the sample data are not perfectly representative of this, or (b) two or more sample means represent different population means.

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It is useful to include the 95% confidence intervals when results are shown as figures because the degree of overlap between them provides an approximate significance of the differences between groups proven 200 mg doxycycline infection 2 game cheats. The interpretation of the degree of overlap is discussed in Chapter 3 (also see Table 3 cheap 100 mg doxycycline antibiotics for acne sun exposure. Many statistics programs do not provide confidence intervals around frequency statis- tics buy 100mg doxycycline with visa antibiotics for uti gram negative. However generic doxycycline 100 mg without prescription bacterial rash, 95% confidence intervals can be easily computed using an Excel spread- √ sheet. The standard error around a proportion is calculated as [p(1–p)∕n] where p is Rates and proportions 259 the proportion expressed as a decimal number and n is the number of cases in the group from which the proportion is calculated. An Excel spreadsheet in which the percentage is entered as its decimal equivalent in the first column and the number in the group is entered in the second column can be used to calculate confidence intervals as shown in Table 8. The formula for the standard error is entered into the formula bar of Excel as sqrt (p × (1 − p)/n) and the formula for the width of the confidence interval is entered as 1. This width, which is the dimension of the 95% confidence interval that is entered into SigmaPlot to draw bar charts with error bars, can then be both subtracted and added to the proportion to calculate the 95% confidence interval values shown in the last two columns of Table 8. The calculations are undertaken in proportions (decimal numbers) but are easily con- verted back to percentages by multiplying by 100, that is, by moving the decimal point two places to the right. Using the converted values, the result could be reported as ‘the percentage of male babies born prematurely was 40. This was significantly different than the percentage of female babies born prematurely which was 20. Because the value of ‘n’ is integral in the denominator of the calculation of confidence intervals, the larger the sample size, the smaller the confidence will be, indicating greater precision in the result. In general, a large sample size is required to reduce 95% confidence intervals below a width of 5%. The lack of overlap between the confidence intervals is an approximate indication of a statistically significant difference between the two groups (see Table 3. Research question Question: Are the babies born in regional centres (away from the hospital or overseas) more likely to be premature than babies born in local areas? Null hypothesis: That the proportion of premature babies in the group born locally is not different to the proportion of premature babies in the groups born regionally or overseas. Variables: Place of birth (categorical, three levels and) prematurity (categorical, two levels) In this research question, there is no clear outcome or explanatory variable because both variables in the analysis are characteristics of the babies. This type of question is asked when it is important to know about the inter-relationships between variables in the data set. If prematurity has an important association with place of birth, this may need to be taken into account in multivariate analyses. The row percentages in the Crosstabulation table show that there is a difference in the frequency of prematurity between babies born at different locations. This difference in percentages fails to reach signifi- cance with a Pearson’s chi-square value of 5. As mentioned previously, Pearson’s chi-square may underestimate the P value when the sample size is small. For tables such as this that are larger than 2 × 2, an Exact chi-square test should be used when an expected count is low (see Section 8. In the crosstabulation, the absolute difference in per cent of premature babies between regional and overseas centres is quite large at 55. In this case, the sample size is too small to demonstrate statistical significance when a large differ- ence of 37. If the sample size had been larger, then the P value for the same between-group difference would be significant. Conversely, the difference between the groups may have been due to chance and a larger sample size might show a smaller between-group difference. The row percentages illustrate the problem that arises when some cells have small numbers. When a group size is small, adding or losing a single case from a cell results in a large change in frequency statistics. Because there are some small group sizes, the footnote in the Chi-Square Tests table indicates that one cell in the table has an expected count less than five. This minimum expected cell count is printed in the footnote below the Chi-Square Tests table. If a table has less than five expected observations in more than 20% of cells, the assumptions for the chi-square test are not met. However, cells and groups with small numbers are a problem in all types of analyses because their summary statistics are often unstable and difficult to interpret. When calculating a chi-square statistic, most packages will give a warning message when the number of expected cases in a cell is low. Pearson’s chi-square tests may be valid when the number of observed counts in a cell is zero as long as the expected number is greater than 5 in 80% of the cells and greater than 1 in all cells. If expected numbers are less than this, then an exact chi-square based on alternative assumptions should be used. The following table is obtained when the Monte Carlo method of computing the exact chi-square is requested. The Monte Carlo P value is based on a random sample of a probability distribution rather than a chi-square distribution which is an approximation. When the Monte Carlo option is selected, the P value will change slightly each time the test is run on the same data set because it is based on a random sample of probabilities. The 264 Chapter 8 two-sided test should be used because the direction of effect could have been either way, that is, the proportion of premature babies could have been higher or lower in any of the groups. An alternative to using exact methods is to merge the group with small cells with another group but only if the theory is valid. It is usually sensible to combine groups when there are less than 10 cases in a cell. Alternatively, the group can be omitted from the analyses although this will reduce the generalizability of the results. As a rule of thumb, the maximum number of cells that can be tested using chi-square is the sample size divided by 10. Thus, a sample size of 160 could theoretically support 16 cells such as an 8 × 2table,a5× 3tableora4× 4 table. However, this relies on an even distribution of cases over the cells, which rarely occurs. In practice, the maximum number of cells is usually the sample size divided by 20. In this data set, this would be 141/20 or approximately seven cells which would support a 2 × 2or2× 3table. The pathway for analyzing categorical variables when some cells have small numbers is shown in Figure 8. However, if two or more unrelated groups need to be combined, they could be described with a generic label such as ‘other’ if neither group is more closely related to one of the other groups in the analysis.

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