Table 5

Dose-Response Effects—Instrumental Variables Estimates, Standardized Outcomes

	Knowledge of Letter Sounds	Reading of Nonwords	Fluency of Oral Reading	Reading Comprehension	Literacy Score
	(1)	(2)	(3)	(4)	(5)
Days in tutorial × End of Grade 3	0.009^***	0.002	0.004^***	0.000	0.007^***
	[0.002]	[0.001]	[0.001]	[0.001]	[0.001]
Days in tutorial × Beginning of Grade 4	0.007^***	0.003^**	0.005^***	0.004^***	0.007^***
	[0.001]	[0.001]	[0.002]	[0.001]	[0.002]
Days in tutorial × End of Grade 4	0.007^***	0.002^**	0.002	0.001	0.004^***
	[0.001]	[0.001]	[0.001]	[0.001]	[0.001]
Observations	4,949	6,362	6,362	6,362	6,362
p-value of equal coeffs.	0.279	0.742	0.143	0.026	0.042
Baseline control mean (non-std.)	15.02	26.94	46.32	3.303	0.507
Baseline control SD (non-std.)	11.36	10.04	13.81	1.877	0.142

Notes: Each column shows the estimates of the coefficients β_h of Equation 3, that is, the estimated dose response at different time horizons for each outcome of interest. Dosage is measured by the number of days in which students attended the tutorial. The actual attendance was instrumented with the randomized treatment indicator variable. The first stage Kleibergen–Paap F-statistic is equal to 1828, and the associated p-value is zero. The row labeled “p-value of equal coeffs.” shows the p-value of a test H₀ : β₁ = β₂ = β₃. Control mean and SD correspond to the number of correct answers (Columns 1–4) or the percent of correct answers (Column 5). All models include cohort, year, and strata fixed effects. Standard errors, shown in brackets, are clustered at the school level (the unit of randomization). Significance: ^*p < 0.10, ^**p < 0.05, ^***p < 0.01.