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Weak instrument bias, a phenomenon by which instrumental-variable estimates using variants not strongly associated with the risk factor of interest are biased, may be substantial if large numbers of genetic variants are used (30), as may be necessary in a multivariable Mendelian randomization experiment.

However, the data-driven choice of instrumental variables can also lead to weak instrument bias [ 34]; hence the analysis using all variants regardless of their association with calcium levels is also performed.

By the end of the 19th century, the alto flute, a weak instrument dynamically, had joined the orchestra, but it was exploited most effectively by the French composers of the early 20th century—for example, Claude Debussy.

The p value of the underidentification test also allows the rejection of the hypothesis of having a weak instrument.

iii) The weak instrument Kleibergen-Paap rank Wald F-statistic to assess if instruments are sufficiently correlated with the right hand side endogenous variable.

Because of a weak instrument problem, only OLS results are shown, which points to a need for caution in interpreting them.

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In two-sample Mendelian randomization, any bias from weak instruments (instrumental variables that are not strongly associated with the risk factor) is in the direction of the null [ 17], so the use of large numbers of genetic variants which are valid instrumental variables should not result in causal claims which are false positives.

It should be noted that, in the specifications based on data aggregated at the municipal level, the instrumental variables used pass the "weak instruments" and "Wu-Hausman" tests.

Moreover, the test for weak instruments reveals that they are jointly significant in the probit specification.

The null hypothesis of weak instruments can be rejected at 5%% of distortion.

However, given these weak instruments, the power of the Wu-Hausman test is acknowledged to be poor, and therefore, we proceed to employ an alternative IV strategy that caters for the weak instruments used.

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