61 Section 1: Statistical Terms and Equations Chapter 6 A Simple Regression Table Introducing the use of Excel to obtain a simple regression table for ability-attainment comparison. In the previous chapter, you learned how to calculate an expected score from one set of ability and attainment scores. The busy assessor needs to be able to work efficiently, not having to perform relatively arduous calculations every time they are involved with their clients. In this case example, we learn how to create a table of regressed scores ranging across the normal distribution of ability for you to save and apply at appropriate times. Case Example 6 You assess three young children, Anne, Suzie and Poppy, all of similar school age, on the KBIT-2 and obtain measures of their composite IQ standard scores. You then administer a test of reading comprehension to the children. You obtain the correlation coefficient for the KBIT-2 and the test of reading comprehension which is 0.54. The standard scores on all the tests are recorded in Table 6.1. Table 6.1: KBIT-2 Composite IQ and Reading Comprehension Scores Composite IQ Reading Comprehension Score Anne 95 105 Suzie 75 80 Poppy 135 100 You want to have available a look-up table of IQ and predicted attainment scores in order to be able to confirm quickly if the scores on the comprehension test are within expectation or not. Points to Consider The previous chapter stated that regression to the mean has more impact for very high or very low scores. Therefore, you would expect that by application of the simple regression formula we used in the previous case example that Anne’s IQ standard score of 95 would lead to a regressed, predicted attainment score not too dissimilar to a score of 95. In contrast, we would expect Poppy’s high IQ standard score of 135 to lead to a significantly regressed attainment score. Likewise, we would expect Suzie’s expected attainment score to be regressed significantly to the mean from the opposite direction of the standard normal distribution curve. The latter case example perhaps helps the assessor faced with a ‘low IQ/but higher attainment level’ case understand that such scenarios are to be expected as a product of the statistical phenomenon of regression to the mean. SAMPLE
Transcript
61Section 1: Statistical Terms and Equations
Chapter 6 A Simple Regression Table
Introducing the use of Exceltoobtainasimpleregressiontableforability-attainmentcomparison.
In the previous chapter, you learned how to calculate an expected score from one set of ability and attainment scores. The busy assessor needs to be able to work efficiently, not having to perform relatively arduous calculations every time they are involved with their clients. In this case example, we learn how to create a table of regressed scores ranging across the normal distribution of ability for you to save and apply at appropriate times.
CaseExample6You assess three young children, Anne, Suzie and Poppy, all of similar school age, on the KBIT-2 and obtain measures of their composite IQ standard scores. You then administer a test of reading comprehension to the children. You obtain the correlation coefficient for the KBIT-2 and the test of reading comprehension which is 0.54. The standard scores on all the tests are recorded in Table 6.1.
Table 6.1: KBIT-2 Composite IQ and Reading Comprehension Scores
Composite IQ Reading Comprehension Score
Anne 95 105
Suzie 75 80
Poppy 135 100
You want to have available a look-up table of IQ and predicted attainment scores in order to be able to confirm quickly if the scores on the comprehension test are within expectation or not.
Points to ConsiderThe previous chapter stated that regression to the mean has more impact for very high or very low scores. Therefore, you would expect that by application of the simple regression formula we used in the previous case example that Anne’s IQ standard score of 95 would lead to a regressed, predicted attainment score not too dissimilar to a score of 95. In contrast, we would expect Poppy’s high IQ standard score of 135 to lead to a significantly regressed attainment score. Likewise, we would expect Suzie’s expected attainment score to be regressed significantly to the mean from the opposite direction of the standard normal distribution curve. The latter case example perhaps helps the assessor faced with a ‘low IQ/but higher attainment level’ case understand that such scenarios are to be expected as a product of the statistical phenomenon of regression to the mean.
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62 Section 1: Statistical Terms and Equations
In fact, the actual regressed scores can be show as in Figure 6.1.
Figure 6.1: Regressed Scores
TheStatisticsExcel provides a means of generating a useful look-up table to use when comparing ability and attainment test scores but where it is anticipated that for each pair of selected ability and attainment tests there will be a particular correlation coefficient and where it should be expected that the correlation coefficients for each pair selected will differ again for the age ranges within these tests.
Figure 6.2 shows a sample from a table designed using Excel, a version of which is available in Appendix E.
Figure 6.2: Predicted Scores for a Range of Correlation Coefficients
Columns A and O give you the IQ scores, starting with an IQ of 139 in row 25 (rows 4-24 have been hidden) and descending to an IQ of 121 in row 43. Row 3 gives a range of correlation coefficients starting with 0.30 in cell B3 and rising in units of 0.5 up to 0.90 in cell N3. Thus by selecting a particular IQ score and confirming the correlation coefficient for the two tests selected, you identify the relevant column from the range B-N. The selected cell for the IQ value will then give you the predicted, regressed attainment score.
Therefore, Poppy’s IQ of 135 with a correlation coefficient of 0.54 for the tests of ability and attainment gives you a predicted, regressed score of 119.
Chapter 6 A Simple Regression Table
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63Section 1: Statistical Terms and Equations
Now obtain the respective regressed scores for Anne and Suzie from Appendix E. Anne’s IQ of 95 gives a regressed score of 97 and Suzie’s IQ of 75 gives a regressed score of 86.
As predicted, and as shown in Figure 6.1, Anne’s regressed score does not differ greatly to her IQ score of 95 because it lies near to the mean of 100. Table 6.1 can be extended as shown in Table 6.2 to give the predicted, regressed scores and the differences between these predicted and observed scores.
Table 6.2: Predicted Reading Comprehension Scores from KBIT-2 Composite IQ Scores
KBIT-2 Composite IQ
Reading Comprehension
Score
Predicted Regressed Score
Difference between Predicted and
Observed Score (p-o)
Anne 95 105 97 +8
Suzie 75 80 86 -6
Poppy 135 100 119 -19
Let us see how the look-up table was created. We do this by examining the formula bar, and with a sample of the table for Poppy’s regressed score of 119 as given below in Figure 6.3.
Figure 6.3: An Example of Looking Up a Regressed Score
Cell G29 is highlighted and, in doing, so the regression formula:
y’ = rxy (X – 100) + 100
that we initially used in Chapter 6 for the creation of the value of 119 is revealed in the formula bar:
If we ignore the $ symbols in the formula for the time being, you can see that G3 cell refers to the correlation coefficient selected. This is multiplied (*) by the total of the ability score given in cell A29 minus 100. 100 is then added. The process can be repeated for the range of the other correlation coefficients in row 3.
$ symbols are used in the formula because these are necessary to freeze the value of certain cells used in the formula when using the ‘fill handle’ facility in Excel to fill other cells.
The following chapters will expand more on how you can advance to the next stage of making decisions about the size of the discrepancies that you are observing as given in the three examples above.
When using Appendix E for correlation coefficients that lie between the values given in the columns, always remember to choose the column for the nearest correlation coefficient.
