A researcher is interested in whether or not children diagnosed with autism differ from other children in the number of digits from a list that they can correctly repeat back to the experimenter. It is known that the average number of digits that children without autism in the population can repeat back is µ = 7.8. A sample of 10 children with autism yielded the scores shown in the table below; the table shows how many digits each participant was able to repeat back to the experimenter.
Q. Continuing with this example, (i.e. the same population mean, etc.), calculate the effect size for the comparison of the autistic and non-autistic children, using Cohen's d. Also, calculate r2 for this problem. How large of an effect size is this, small, medium, or large
Full Answer Section
Interpretation of Cohen's d:
- 0.2: Small effect size
- 0.5: Medium effect size
- 0.8: Large effect size
R-squared (coefficient of determination):
This statistic represents the proportion of variance in the dependent variable (digit span) explained by the independent variable (autism diagnosis).
Calculation (assuming data is available):
- Requires calculating the explained variance (regression) and total variance.
Interpretation of R-squared:
- Values closer to 1 indicate a larger proportion of variance explained by autism diagnosis.
Next Steps:
If you have the individual digit span scores for the autistic children, you can calculate the sample mean and standard deviation for their group. Then, plug those values along with the population mean for non-autistic children (µ = 7.8) into the Cohen's d formula.
For r-squared, you would need to conduct a statistical test like linear regression to determine the explained variance and calculate the total variance.
With the effect size (Cohen's d) value, you can determine if there's a small, medium, or large difference in digit span between autistic and non-autistic children in this sample.
Sample Answer
Effect Size and Interpretation for Autism and Digit Span Task
Given Information:
- Children with Autism (Sample): n = 10
- Population Mean (Non-Autistic): µ = 7.8 digits
- We don't have the individual digit span scores for the autistic children.
Limitations:
Due to missing data on individual digit span scores for the autistic children, we cannot calculate the sample mean (average score) for the autistic group or perform the specific calculations for Cohen's d and r-squared.
However, we can explain the concepts and how to proceed if the data were available:
Cohen's d:
This effect size statistic measures the difference between the means of two groups in standard deviation units.
Formula (assuming data is available):
d = (Mean_Autistic - µ) / Standard Deviation_Autistic
- Mean_Autistic: Average digit span score for the autistic children (needs data).
- µ: Population mean for non-autistic children (given as 7.8).
- Standard Deviation_Autistic: Variability of digit span scores in the autistic group (needs data collection and calculation).