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Research Article

Cognitive fatigue influences students’ performance on standardized tests

Hans Henrik Sievertsen, Francesca Gino, and Marco Piovesan
PNAS March 8, 2016 113 (10) 2621-2624; first published February 16, 2016; https://doi.org/10.1073/pnas.1516947113
Hans Henrik Sievertsen
aThe Danish National Centre for Social Research, 1052 Copenhagen, Denmark;
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Francesca Gino
bHarvard Business School, Harvard University, Boston, MA 02163;
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  • For correspondence: fgino@hbs.edu
Marco Piovesan
cDepartment of Economics, University of Copenhagen, 1353 Copenhagen, Denmark
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  1. Edited by Pamela Davis-Kean, University of Michigan, Ann Arbor, MI, and accepted by the Editorial Board January 15, 2016 (received for review August 25, 2015)

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  • Fig. 1.
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    Fig. 1.

    Hour-to-hour effect on test scores and break patterns. Effects are estimated based on administrative data from Statistics Denmark. (Upper) How the average test score changes from hour to hour. (Lower) Distribution of when breaks end, based on a survey conducted on 10% of the schools. The hourly effect is estimated in a linear model controlling for unobserved time invariant fixed effects on grade, day of the week, and school level. We also control for test year fixed effects, as well as parental income, parental education, nonwestern origin, sex, spring child, and birth weight. The details on the model and estimation procedure are shown in SI Text, along with a table with regression results.

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    Fig. 2.

    Effect of time of day and breaks. Effects are estimated based on administrative data from Statistics Denmark. The figures show the parameter estimates for break and test hour from estimating a linear model of test score on test hour and break and controlling for test year fixed effects, as well as parental income, parental education, nonwestern origin, sex, spring child, and birth weight. We also control for school, grade, subject, and day of the week fixed effects. The details on the model and estimation procedure are shown in SI Text, along with a table with regression results. (A) Main effect and the effect by subgroups. (B) Results from quantile regression at the 10th, 20th, 30th, 40th,…,90th percentiles using Canay’s plugin fixed effect estimator (21). The graph shows the effect of breaks and test hour over the test score distribution.

  • Fig. S1.
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    Fig. S1.

    Test score distribution. The plots were created using a triangular kernel with a bandwidth of 0.25, separately for each test hour.

  • Fig. S2.
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    Fig. S2.

    z-Scores for different specification of model S1 and varying sample size. The t-values are calculated by estimating model S2 with fixed effects, but without any individual controls, using OLS. The t-values are then obtained by t=α^1/se^(α^1). The green scatters are obtained from models using the test scores as dependent variables. The gray scatters are obtained from estimating the model using the corresponding covariate as the dependent variable.

Tables

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    Table S1.

    Main regression results

    Variable1234567
    testhour−0.006***−0.005***−0.008***−0.009***
    (0.001)(0.001)(0.001)(0.001)
    break0.025***0.019***0.017***
    (0.006)(0.003)(0.003)
    9−0.050***−0.038***−0.035***
    (0.009)(0.004)(0.004)
    10−0.038***−0.029***−0.028***
    (0.005)(0.003)(0.003)
    11−0.060***−0.055***−0.052***
    (0.009)(0.004)(0.004)
    12−0.033***−0.042***−0.042***
    (0.007)(0.004)(0.004)
    13−0.039***−0.052***−0.054***
    (0.011)(0.006)(0.006)
    Fixed effects includedNoNoYesYesNoYesYes
    Individual controlsNoNoNoYesNoNoYes
    Number of schools2,0282,0282,0282,028
    Smallest group2222
    Largest group3,9843,9843,9843,984
    F-value th9 = th10..= th13 = 014.8438.6139.59
    P value th9 = th10..= th13 = 00.000.000.00
    Model degrees of freedom5203021727
    Adjusted R20.000.000.080.000.000.08
    AIC5,774,7565,774,1885,626,9055,462,8865,774,5775,627,2915,463,303
    Observations2,034,9642,034,9642,034,8872,034,8872,034,9642,034,8872,034,887
    • The dependent variable in each model is standardized test score. All estimates are obtained using ordinary least squares (OLS). Column 1 shows the point estimate for α1 from estimating model S1. Columns 2–4 show results from estimating model S2. Columns 5–7 show results from estimating model S3. The table only shows the point estimates for the coefficients α1−α5 in model S2 and coefficients α1 and α2 for model S3. Columns 2 and 5 show results from estimating models without any control variables. Columns 3 and 6 show results from estimating simple models with only school, year, day of the week, grade, and subject fixed effects. Columns 4 and 7 show results from estimating the full models without individual fixed effects. SEs clustered at the school level are shown in parentheses. Number of schools show the number of schools included and thus also the level of fixed effects and clustering. Smallest/largest group shows the smallest/largest number of observations from one school. F-value gives the F-statistic for a test of joint significance for the hourly indicators in columns 2–5, and P value gives the corresponding P values. This P value is for the null hypothesis that all hourly indicators are zero. Model degrees of freedom specifies the number the degrees of freedom used by the model. AIC gives the Akaike information criteria. Smaller AICs are generally preferred. The term observations refers to the number of observations included in the regressions. The dependent variable is standardized within the test year, grade, and subject cell. As fixed effects on the school level implies comparisons within schools, we only include schools with at least two tests. Regressions are based on administrative data from Statistics Denmark and the Danish Ministry for Education, for all mandatory tests 2009/10–2012/13.

    • ↵*** P < 0.01; **P < 0.05; *P < 0.1.

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    Table S2.

    Variable means across test time

    VariableAllTest hour
    8910111213
    Full sample
     Test uncertainty0.260.260.270.260.280.260.26
     School day179.89180.10180.02178.81180.49180.83179.80
     Child birth weight (g)3,2983,3073,3043,3003,3033,2823,272
     Parents’ years of schooling14.2414.2614.1814.2114.2514.3014.33
     Household income, 1,000 DKK386.17384.64379.79386.07382.60396.16396.96
     Household income percentile57.0557.1456.3256.9856.7757.9158.16
     Nonwestern origin0.090.080.090.090.080.090.10
     Female0.490.490.480.490.490.490.49
     Spring child0.500.500.490.490.500.500.50
     Missing birth weight data0.060.060.060.060.060.060.06
     Missing education data0.030.030.030.030.020.020.01
     Observations2,034,964400,139424,868528,764259,649295,032126,512
    Subsample of schools included in the break data sample
     Test uncertainty0.240.240.220.250.270.240.25
     School day179.65179.18179.57180.18179.61179.86178.93
     Child birth weight (g)3,3053,3023,3033,3033,3213,2953,323
     Parents’ years of schooling14.2914.2614.1714.3014.3514.3814.40
     Household income, 1,000 DKK389.37392.55376.89393.87385.21397.78390.23
     Household income percentile57.2557.4356.1157.4257.0658.1058.18
     Nonwestern origin0.080.080.080.090.060.080.08
     Female0.490.500.490.490.490.490.48
     Spring child0.500.490.500.490.490.500.51
     Missing birth weight data0.060.060.060.060.050.060.05
     Missing education data0.030.030.030.030.020.020.01
     Observations121,70925,56124,02531,19817,03916,9956,891
    • The table shows simple averages for subsamples across test hours. Test uncertainty is the estimated uncertainty for the test result (only available for years 2011–2013). This variable measures how precisely the individual test score was estimated by the computer. (The uncertainty comes from the fact that the test can be more or less accurate to measure the student performance. Each question contributes to the precision of the test, reducing uncertainty. That is, the test is adaptive, and the computer tries to calculate the individuals level. For instance, if the student reaches a level where every time he/she gets a harder question, he/she answers wrong, but every time he/she gets an easier question, he/she answers right, there will be very low uncertainty on test score.) Income is adjusted to the 2010 price level and adjusted for household size using the square root approach. Spring child is a child born in the period January–June of the year being considered. Because the school starting age cutoff is January 1, these children have the highest expected school starting age. The variable called school days refers to the number of days from the start of the school year to the test day, not counting weekends. Parents’ years of schooling is the years of schooling completed by the mother or father (the highest value).

    • View popup
    Table S3.

    Regression results

    VariableMain (1)Math (2)Reading (3)Young (4)Old (5)Break data (6)Individual FE (7)
    testhour−0.009***−0.009***−0.003**−0.006***−0.010***−0.005−0.011***
    (0.001)(0.002)(0.001)(0.002)(0.001)(0.003)(0.001)
    break0.017***0.018**0.017***0.013**0.016***0.027**0.008***
    (0.003)(0.006)(0.004)(0.005)(0.003)(0.009)(0.002)
    Number of schools/individuals2,0281,9012,0151,8681,98784492,020
    Smallest group21111952
    Largest group3,9848351,5521,2853,1303,19611
    Model degrees of freedom27192120242726
    Adjusted R20.0780.0810.0970.0830.0770.0810.002
    AIC5,462,8091,135,2882,220,0671,704,0953,742,861323,2083,139,887
    Observations2,034,887426,615836,115637,9741,396,913121,7091,956,608
    • Dependent variable: standardized test scores. The row break shows the point estimate for the coefficients on break and test hour in model S3. All models are estimated with the full set of covariates and fixed effects. Column 1 is for the full sample. In column 2, only tests in mathematics are included. In column 3, only tests in reading are included. In column 4, only tests for grades 2–4 are included. In column 5, only tests for grades 6–8 are included. In column 6, we only included the schools included in the break survey. In column 7, we estimated a modified version of model S2, using individual fixed effects. SEs clustered at the school level are shown in parentheses. Number of schools show the number of schools included (individuals for column 7, and thus also the level of fixed effects and clustering (Individual FE in column 7 is also clustered at the school level). Smallest/largest group shows the smallest/largest number of observations from one school/one individual. Model degrees of freedom specifies the number the degrees of freedom used by the model. AIC gives the Akaike information criteria. Smaller AICs are generally preferred. Observations refers to the number of observations included in the regressions. The dependent variable is standardized within the test year, grade, and subject cell. As fixed effects on the school level implies comparisons within schools, we only include schools with at least two tests. Regressions are based on administrative data from Statistics Denmark and the Danish Ministry for Education, for all mandatory tests 2009/10–2012/13.

    • ↵*** P < 0.01; **P < 0.05; *P < 0.1.

    • View popup
    Table S4.

    Regression results

    Variable1234567
    testhour−0.159***−0.127***−0.226***−0.236***
    (0.038)(0.036)(0.023)(0.022)
    break0.691***0.534***0.487***
    (0.188)(0.075)(0.071)
    9−1.353***−1.069***−0.978***
    (0.247)(0.111)(0.106)
    10−0.983***−0.788***−0.749***
    (0.149)(0.094)(0.090)
    11−1.556***−1.470***−1.392***
    (0.250)(0.123)(0.118)
    12−0.844***−1.130***−1.124***
    (0.190)(0.114)(0.110)
    13−1.077***−1.463***−1.524***
    (0.305)(0.165)(0.157)
    Fixed effects includedNoNoYesYesNoYesYes
    Individual controlsNoNoNoYesNoNoYes
    Number of schools2,0282,0282,0282,028
    Smallest group2222
    Largest group3,9843,9843,9843,984
    F-value th9 = th10..= th13 = 011.71136.61737.727
    P value th9 = th10..= th13 = 00.0000.0000.000
    Model degrees of freedom5203021727
    Adjusted R20.000.000.080.000.000.08
    AIC19,460,40119,459,91319,320,48819,149,059194,601,2919,3205,4419,149,447
    Observations2,034,9642,034,9642,034,8872,034,8872,034,9642,034,8872,034,887
    • Dependent variable: test score percentiles. All estimates are obtained using OLS. Column 1 shows the point estimate for α1 from estimating model S1. Columns 2–4 show results from estimating model S2 using ordinary least squares. Columns 5–7 show results from estimating model S3 using ordinary least squares. The table only shows the point estimates for the coefficients α1−α5 in model S2 and coefficient α1 and α2 for model S3. Columns 2 and 5 show results from estimating models without any control variables. Columns 3 and 6 show results from estimating simple models with only school, year, day of the week, grade, and subject fixed effects. Columns 4 and 7 show results from estimating the full models without individual fixed effects. SEs clustered at the school level are shown in parentheses. Number of schools show the number of schools included and thus also the level of fixed effects and clustering. Smallest/largest group shows the smallest/largest number of observations from one school. F-value gives the F-statistic for a test of joint significance for the hourly indicators, and P value gives the corresponding P values. Model degrees of freedom specifies the number the degrees of freedom used by the model. AIC gives the Akaike information criteria. Smaller AICs are generally preferred. Observations refers to the number of observations included in the regressions. The dependent variable is test score percentile rank (1–100) within the test year, grade, and subject cell. As fixed effects on the school level implies comparisons within schools, we only include schools with at least two tests. Regressions are based on administrative data from Statistics Denmark and the Danish Ministry for Education, for all mandatory tests 2009/10–2012/13.

    • ↵*** P < 0.01; **P < 0.05; *P < 0.1.

    • View popup
    Table S5.

    Regression results

    Variable1234
    Household income0.0013***
    (0.0000)
    Birth weight0.0001***
    (0.0000)
    Years of schooling0.1195***
    (0.0010)
    School days0.0009***
    (0.0001)
    Adjusted R20.060.000.100.00
    AIC5,554,8285,361,7765,404,7955,773,793
    N2,002,0331,899,1591,981,6712,034,964
    • Dependent variable: standardized test score. SEs clustered at the school level are shown in parentheses. AIC gives the Akaike information criteria. Observations shows number of observations included in the regressions. The dependent variable is standardized within the test year, grade, and subject cell. We removed outliers in income and birth weight (first and last percentile). This is done because, in this linear regression, measurement errors (e.g., birth weight of 10 kg) would have a huge impact on the point estimates. However, overall the conclusions are not very sensitive to this change.

    • ↵*** P < 0.01; **P < 0.05; *P < 0.1.

Data supplements

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Cognitive fatigue in school
Hans Henrik Sievertsen, Francesca Gino, Marco Piovesan
Proceedings of the National Academy of Sciences Mar 2016, 113 (10) 2621-2624; DOI: 10.1073/pnas.1516947113

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Cognitive fatigue in school
Hans Henrik Sievertsen, Francesca Gino, Marco Piovesan
Proceedings of the National Academy of Sciences Mar 2016, 113 (10) 2621-2624; DOI: 10.1073/pnas.1516947113
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