# Recommend Continued CPS Monitoring. 63 (a) 17 (b) 10 (c) (d) 20 (e) 25 (f) 80. Totals/Marginal

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1 Work Sheet 2: Calculating a Chi Square Table 1: Substance Abuse Level by ation Total/Marginal 63 (a) 17 (b) 10 (c) (d) 20 (e) 25 (f) 80 Totals/Marginal Step 1: Label Your Table. Label the 3x2 table (, Continues, BY with AND ) cells with a letter (i.e., 63 [a], 17 [b], 10 [c], then move to 2 nd row, 35 [d], 20 [e], 25 [f]). Step 2: IV and DV. Identify the IV(s) and the DV(s). IVs are rows and DVs are columns. IV is the whether parents are involved with substances. DV is the type of recommendation made by CPS. Step 3: Hypothesis. Develop the research and null hypotheses. What is your research hypothesis? o There is a relationship between the level of involvement with substances and the recommendations made by CPS. What is the null hypothesis? o There is not a relationship between the level of involvement with substances and the recommendations made by CPS. Page 1 of 6

2 Step 4: Complete a Dummy Table. A dummy table is a visual depiction of where large portions of the observations should fall if there is a relationship between the variables. Put asterisks in the cells where you would predict the outcomes to be. Table 2: Dummy Table for Substance abuse and ation * * * * Step 5: Calculate a Percentage Table. Use a percentage table to see the differences in the frequencies (to create a percentage, divide the number of observations in each cell by the total number of observations, in cell A, 63/90 =.7 = 70%). Complete the percentage table. Table 3: Percentages of Clients: Substance abuse history by ation Total Percentage 70% 18.9% 11.1% 100.0% 43.75% 25% 31.25% 100.0% Page 2 of 6

3 HOWEVER, THE TABLES DOES NOT TELL YOU WHETHER THESE DIFFERENCES IN PERCENTAGES ARE STATISTICALLY SIGNIFICANT. Step 6: Calculate Expected Frequencies. These are the frequencies you would expect to occur most often in an infinite number of samples if the null hypothesis is correct that is, there is NO true relationship between the two nominal level variables. These are a hypothesized number based on a mathematical formula. Expected cell frequency is: ( )( ) E = expected frequency R = Total number in that cell s row C = Total number in that cell s column Calculate the expected frequencies for each of the cells in our contingency table Table 4: Observed and Expected Frequencies Total/Marginal Obs. Expected Obs. Expected Obs. Expected Totals/Marginal Page 3 of 6

4 Step 7: Calculate the 2. Need to calculate the differences for each cell between the observed and expected frequencies divided by the expected frequencies for those cells. These are then totaled together to determine the Chi-Square estimate. = ( ) (for all cells) O= Observed frequency E= Expected frequency 2 = cell a + cell b + cell c + cell d + etc. Cell a b c d e f 2= ( ) + ( ) + ( ) + ( ) + ( ) + ( ) = = Step 8: Degrees of Freedom. Degrees of freedom represent the number of ways that a system can vary in either direction without violating the assumptions placed upon the system. The degrees of freedom are determined by the number of bits of information fed into the system. To determine the degrees of freedom for your table, multiply the (number of row variables minus 1) by the (number of column variables minus 1). df = (r-1)(c-1) r = number of rows; c = number of columns df = (2-1)(3-1) = (1)(2) = 2 Page 4 of 6

5 Step 9: Use df to determine critical value of 2 Use critical value table to determine critical χ 2 for the different probability levels (Google critical value table for chisq ) Using the df you estimated, go down the column on the far left and across the row to find the critical χ 2 value. When the critical χ 2 statistic you computed from the observed data exceeds the critical value in the table for at least a 0.05 probability level (i.e., = 0.95), then we reject null hypothesis. What is the computed χ 2 value? What is the critical value of the χ 2 = 5.99? Based upon these values, should you reject the null hypothesis (that there is no relation between the observed and statistically expected outcomes)? Here the computed value of the χ 2 statistic is 14.13, which is above the critical value at the.05 level. We can reject the null. In fact, on the table, it is beyond the.001 significance level. (Go along the row until find the calculated value and where it falls in terms of significance.) Step 10: Interpreting the findings. If significant: Tells you that the differences between the observed data and what would be expected based upon your values were not observed by chance. In other words, the relationship between the values observed is significant, but this alone will not allow us to assume a causal relationship. Can now go back to dummy table and see if the differences between observed and expected were as predicted. Look for disproportionately large frequencies (a disproportionately larger observed frequency is one that is greater than the expected frequency). You want to see if your findings support your hypothesis. They may not. There may be differences, but they may be between variables that were different than predicted. Before one can claim an identified relationship between variables would need to rule out other alternative explanations (this is the purpose of a research design, Module II). The most you can say is that there is a statistically significant relationship between the variables, but cannot say why, or that these results are generalizable to other samples. To do this, you need further evaluation to examine if the pattern persists. Page 5 of 6

6 AGAIN FOR COMPARISON Table 4: Observed and Expected Frequencies Total/Marginal Obs. Expected Obs. Expected Obs. Expected Totals/Marginal Highlighted sections show differences between expected and observed frequencies. Step 11: Report and interpret your findings. When reporting a chi square, or any statistic, it is important to interpret the findings. To do this, use the percentages or numbers observed contrasted with those expected. Put the findings in context before telling readers that the comparison was significant. For the statistics, provide the following chi-square details: χ 2 =, df =, p <. You may also present your contingency table to show readers the differences. You can report the results of the example here in the following manner: Among 170 CPS cases examined of caregivers with substance abuse involvement, more cases (n=25) were terminated than statistically expected and fewer cases (n = 35) were recommended for reunification. Among caregivers without substance abuse involvement, more cases (n = 63) were reunified than expected whereas fewer cases (n=10) than expected were recommended for termination of parent-child relations. When Comparing the recommendations of CPS workers (reunify, continue monitoring, terminate parent-child relations) with the parent substance abuse behavior, we see a statistically significant relationship between these conditions (χ 2 =14.13, df = 2, p <.0001). Page 6 of 6

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