Estimate variance in a nested study designI have a "mother" bacteria. I cloned it R times and measured a variable P on these R clones and recorded the mean p¯ and the variance V among these clones.Then, I made M mutants (M new bacterias each carrying a single new mutation) from the "mother bacteria". I cloned each of these M mutants r times and measured the variable P again. I recorded the mean Pm and variance Vm of each mutant m (1≤m≤M).The parameter I would like to estimate is the the mean square differences between the mean value of the "mother bacteria" (estimated as p¯) and the mean values of the M mutants (estimated as P¯m).∑Mm(P¯−P¯m)2M would be a biased estimator as I have to remove the variance caused by imprecised estimation of both P¯m and P¯. I think the solution should be along the lines of∑Mm(P¯−P¯m)2M−∑MmVmr−VR

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Estimate variance in a nested study designI have a &quot;mother&quot; bacteria. I cloned it R times and measured a variable P on these R clones and recorded the mean p¯ and the variance V among these clones.Then, I made M mutants (M new bacterias each carrying a single new mutation) from the &quot;mother bacteria&quot;. I cloned each of these M mutants r times and measured the variable P again. I recorded the mean Pm and variance Vm of each mutant m (1≤m≤M).The parameter I would like to estimate is the the mean square differences between the mean value of the &quot;mother bacteria&quot; (estimated as p¯) and the mean values of the M mutants (estimated as P¯m).∑Mm(P¯−P¯m)2M would be a biased estimator as I have to remove the variance caused by imprecised estimation of both P¯m and P¯. I think the solution should be along the lines of∑Mm(P¯−P¯m)2M−∑MmVmr−VR

Kody Arellano · Accepted Answer

Let&#039;s say the true means are   μ and       μ    m   for m=1,…,M for the mother bacteria and the       m          th       mutation, respectively. You want to estimate            ∑              m        =        1            M        (          μ      m        −    μ          )      2        =          ∑              m        =        1            M              μ      m      2        −    2    μ          ∑              m        =        1            M              μ      i        +    M          μ      2        .  Let&#039;s now try to come up with unbiased estimators of each of the terms. By linearity of expectation, we will be done. Note that for any random variable   Var  (  X  )  =      E    (      X    2    )  −  (      E    (  X  )      )    2    .. With this in mind we define  V  =            1              R        −        1                        ∑              i        =        1            R        (          P      i        −                  P        ¯                    )      2             and               V      m        =                  1                  R          −          1                                    ∑                  i          =          1                R            (              P                  m          ,          i                    −                                    P            ¯                          m                    )        2            where       P    i   is the       i          th       measurement from the mother bacteria,   i  =  1  ,  …  ,  R and       P          m      ,      i       is the       i          th       measurement from the       m          th       mutant,   m  =  1  ,  …  ,  M and   i  =  1  ,  …  ,  r  .. Also,             P      ¯        =            1      R                  ∑              i        =        1            R              P      i       and                     P        ¯              m    =            1      r                  ∑              i        =        1            r              P              m        ,        i              .  Then       E    (  V  )  =  Var  (      P          mother        )  ,, and if we define the estimator   T  =            1      R                  ∑              i        =        1            R              P      i      2        −    V    ,  , we will get       E    (  T  )  =      E    (      P    mother    2    )  −  Var  (      P    mother    )  =      E    (      P    mother        )    2    =      μ    2    ..Similarly define       T    m    =            1      r                  ∑              i        =        1            r              P              m        ,        i            2        −          V      m       to get       E    (      T    m    )  =      μ    m    2    .. We then define the estimator      T          ∗        =            ∑              m        =        1            M              T      m      2        −    2                  P        ¯                    ∑              m        =        1            M                                P          ¯                    m        +    M    T    .

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