Question

ADHD? OR JUST YOUNGEST IN THE CLASS?

A study¹ indicates that the youngest children in a school grade are more likely to be diagnosed with attention-deficit/hyperactivity disorder (ADHD) than their older peers in the same grade. The study involved 937,943children between 6 and 12 years old in British Columbia, Canada. The cutoff date for entering school in any year in British Columbia is December 31^st, so in any given class, those born late in the year are almost a year younger than those born early in the year. Is it possible that the younger students are being over-diagnosed with ADHD?
Boys: ADHD or Just Young?
The table below shows the number of boys diagnosed with ADHD based on the quarter of the year in which they were born, as well as the proportion of boys born during that quarter.

Birth Date	ADHD Diagnoses	Proportion of Births
Jan-Mar	6880	0.244
Apr-Jun	7982	0.258
Jul-Sep	9161	0.257
Oct-Dec	8945	0.241

Table 1 ADHD diagnoses and birth date for boys
¹Morrow, R., et al., “Influence of relative age on diagnosis and treatment of attention-deficit/hyperactivity disorder in children,” Canadian Medical Association Journal, April 17, 2012; 184(7): 755–762.

(a) What is the total number of boys diagnosed with ADHD in the sample?
Total number of boys = _____________________

(b) For the null hypothesis, use the overall proportion of births in a quarter to give the null proportion for that quarter. Compute the expected number of ADHD diagnoses for each quarter under this hypothesis.

Round your answers to one decimal place.

Jan-Mar =  _____________

Apr-Jun =   _____________

Jul-Sep =   _____________

Oct-Dec =  _____________

(c) Compute the χ2-statistic.

Round your answer to one decimal place.

χ2 =  _____________

(d) Give the degrees of freedom and find the p-value.

Enter the exact answer for the degrees of freedom, and round your answer for the p-value to three decimal places.

d⁢f=  _____________

p-value =  _____________

(e) State the conclusion of the test.

There is very strong evidence that ADHD diagnoses are related to relative age.

There is some evidence that ADHD diagnoses are related to relative age.

ADHD diagnoses do not seem to be related to relative age in school.

(f) For which group of children does ADHD appear to be diagnosed more frequently than we would expect? Select the group that provides the greatest indication of this.

Jan-Mar

Apr-Jun

Jul-Sep

Oct-Dec

(g) For which group of children does ADHD appear to be diagnosed less frequently than we would expect? Select the group that provides the greatest indication of this.

Jan-Mar

Apr-Jun

Jul-Sep

Oct-Dec

Answer 1

a) total number of boys diagnosed with ADHD in the sample = 6880+7982+9161+8945 = 32968

b) Expected frequencies

Birth Date	ADHD Diagnoses	Proportion of Births (p)	EXPECTED = p*32968
Jan-Mar	6880	0.244	8044.2
Apr-Jun	7982	0.258	8505.7
Jul-Sep	9161	0.257	8472.8
Oct-Dec	8945	0.241	7945.3
Total	32968	1

c) Compute the χ2-statistic.

	Observed	Expected	Difference	Difference Sq.	Diff. Sq. / Exp Fr.
Jan-Mar	6880	8044.2	-1164.20	1355361.64	168.49
Apr-Jun	7982	8505.7	-523.70	274261.69	32.24
Jul-Sep	9161	8472.8	688.20	473619.24	55.90
Oct-Dec	8945	7945.3	999.70	999400.09	125.79

χ2-statistic. = 168.49+32.24+55.90+125.79 = 382.4

d) Degrees of freedom = number of quarters -1 = 4-1 = 3

p-value is < .0001

e) Since p-value is lessthan alpha 0.05 we reject null hypothesis and conclude that There is very strong evidence that ADHD diagnoses are related to relative age.

f) Oct-Dec group of children does ADHD appear to be diagnosed more frequently than we would expect.

g) Jan-Mar group of children does ADHD appear to be diagnosed less frequently than we would expect