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From what I understand, the null hypothesis for One-way ANOVA is:

  1. There is no difference in the means of factor A

The null hypothesis for Two-way ANOVA is:

  1. There is no difference in the means of factor A
  2. There is no difference in means of factor B
  3. There is no interaction between factors A and B

If so, why do I get different F-value for the factor "Country" ? Different F-values [Sry can't embed images due to lack of reputation)

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Because in the two-way anova, you're controlling for the second variable. If the factors are uncorrelated (a balanced design), this will leave your parameter estimates alone, and alter your standard errors (which will change F). If the factors are correlated, this will change your estimates AND standard errors.

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