Causality between two variables | Three conditions for a causality | Possible causality | ||

(Condition 1) Bivariate analysis: Crude covariational relationship | (Condition 2) Multivariate analysis: Covariational relationship through controlling for third variables | (Condition 3) Temporal precedence | ||

Causal path A: on mathematical proficiency
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(Step 1) From not being stunted to attendance rate | YesP=0.003*Mann-Whitney U test* | YesP=0.041*Ordinal regression¶ | Yes While stunting is chronic undernutrition that occurs generally before 3 years of age,24 school attendance takes place after enrolment at 5 or 6 years of age at primary schools. Thus, clearly, school attendance follows becoming stunted. | Yes |

(Step 2) From attendance rate to mathematical proficiency | YesP=0.013*Spearman’s correlation† | YesP=0.014*Ordinal regression‡ | Yes Without attending schools, children cannot take mathematics examinations. Thus, clearly, mathematical proficiency follows attendance. | Yes |

Causal path B: on national language proficiency
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(Step 1) From not being thin to attendance rate | NoP=0.141Mann-Whitney U test* | YesP=0.020*Ordinal regression¶ | Not necessarily Thinness is acute undernutrition that can occur after attendance at school, too. Thus, school attendance does not necessarily follow becoming thin. | No |

(Step 2) From attendance rate to national language proficiency | NoP=0.104Spearman’s correlation (r _{s})† | NoP=0.483Ordinal regression§ | Yes Without attending schools, children cannot take national language examinations. Thus, national language proficiency follows attendance. | No |

*See figure 2.

†See figure 3.

‡Coefficient estimate for not being stunted in multiple regression model with attendance rate (%) and other all background variables as dependent and independent variables, respectively.

§Coefficient estimate for not being thin in multiple regression model with attendance rate (%) and other all background variables as dependent and independent variables, respectively.

¶See table 2.

** *

*P*<0.05