## **_c. Adjustment factor_**

Although the initial sampling approach allows us to run separate analysis for each type of school, when running statistical analyses for the sample as a whole it is necessary to correct the biases introduced with the sampling and to adjust the sample according to the real probability of selection. This was done using an adjustment factor (AF), calculated from a representation factor (RF) for each sample.

**Table 21. Representation and Adjustment Factors by type of school**

|||**Single-**<br>**Teacher**|**Multigrade**|**Fullgrade**|**Total **|
|---|---|---|---|---|---|
|**Number of**<br>**d**|Sample A|120|285|596|**1,001**|
|**stuents**|Sample B|104|164|207|**475**|
||Population|3,157|<br>12,582|16,961|**32,700**|
|**Percentage of**<br>**Sd**|Sample A|12%|28%|60%|**100%**|
|**tuents**|Sample B|22%|35%|44%|**100%**|
||Population|10%|38%|52%|**100%**|
|**Representation**|Sample A|26|44|28|**99**|
|**Factor**|<br>Sample B|30|77|82|**189**|
|**Adjustment**<br>|Sample A|**27%**|**45%**|**29%**|**100%**|
|**Factor**|Sample B|**16%**|**41%**|**43%**|**100%**|

Since students were divided into three groups based on the type of school they attend for the selection process, the RF of each student in the sample is equivalent to the total number of students in his or her group in the population, divided by the total number of students in his or her group in the sample. The AF is calculated from the RF of each group, as a percentage of total RF, which is the sum of the RFs of all three groups (see Table 21). For example, the RF of Sample A for students in single-teacher schools is 3,157 (the total number of students in single-teacher schools in the sample) divided by 120 (the total number of students in single-teacher schools in the population), which results in 26. Then, the AF is 26 (the RF of single-teacher schools) divided by 99 (the sum of all RFs in this sample), multiplied by 100 (to have it as percentage).

# **7.5. Psychometric analysis of the ECE test with Raw vs. Transformed Rasch scores**

To identify the trade-offs of using raw vs. transformed scores, two types of psychometric analyses were run: one for the items and one for the test as a whole. For evaluating the items we used the classic item analysis with raw scores, and the analysis under the Rasch model with transformed scores. The former will be referred to as analysis for raw scores and the latter as analysis for Rasch scores. For evaluating the tests, we used unidimensionality, reliability and normality analysis. This analysis was done with sample A, which includes 1001 cases, using the adjustment factor to match the proportion of students in each type of school to the proportions found in the population.

## **_a. Calculating Raw and Rasch scores_**

For the written test, two sets of scores are calculated: raw and transformed. The raw total score is the sum of correct answers, that is, one point for each question answered correctly. The test contains four pairs of connected questions, namely items 2 and 3, 4 and 5, 6 and 7, and 8 and 9.<sup>52</sup>

> 52 These items are two questions with the same three alternatives of answer, which means that answering one question reduces the chances of error on the other item.

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