power analysis for multilevel model with significant finding

I am working on a study where data collection is still in progress. I have done preliminary analysis with a multilevel model on a small number of people (n=19), and there is a statistically significant treatment effect. There is interest in doing a power analysis now to determine how many additional people to collect data from. Note- I realize power analysis should have been done before data collection began but that wasn’t done in this case. I’m wondering is there any point in doing a power analysis if the results are already statistically significant? If there is, what is the point? 19 seems like a small total number.

Reducing the Imperial strategy’s power in Twilight Imperium

I’ve played Twilight Imperium (3rd edition) a couple times with friends now, and we keep noticing the same thing: no amount of fancy maneuvering, planet conquest, etc. gets people enough victory points to overcome the Imperial strategy card:

Draw the top card from the Objective Deck and place it face up in the common play area. Then receive 2 victory points.

The 2 points from that card are massive, to the point where you can’t afford to choose a different strategy card if that one is in the deck on your turn, or you almost certainly won’t end up winning the game. Is there a way to deal with that? Are there any house rules that weaken it without messing up the game (e.g. only receiving 1 VP)? Or modifying some other part of the game such that the 2 VPs from Imperial aren’t as game-deciding?

Power Query – Value.Divide(Value.Add( )) returns all “null” results

When I create a calculated field in Microsoft Power Query feature (“Get and Transform” since Excel 2016) I use a combination two simple formulae: Value.Divide() and Value.Add(). For some reason, this does not work and all I get is null values.

Here is the series of steps that I do:

  1. Get&Transform data from .csv;
  2. Arrange fields (columns), rename;
  3. Add
    some custom ones;

The one giving me problems is as follows:

= Table.AddColumn(#"Add Cost per result", "ROAS", each Value.Divide(Value.Add([Website conversion value], [#"Mobile app purchases conversion value (corr.)"]), [#"Amount spent (GBP)"]))

Or, as copied from a “graphical interface”:

= Value.Divide(Value.Add([Website conversion value], [#"Mobile app purchases conversion value (corr.)"]), [#"Amount spent (GBP)"])

  • [Website conversion value] is imported from the original .csv;
  • [#”Mobile app purchases conversion value (corr.)”] is calculated field (“Add column…”);
  • [#”Amount spent (GBP)”] is imported from the original .csv, I suspect that this might be the problem one – why does it have a # prepended when it was there originally during the import? Isn’t this a symbol for a “table”?;

Many thanks!

How to determine the change in proportion you would derive from a power calculation

I should start this question by saying that I am not a statistician- so please pardon me if this a very stupid question. I am trying to estimate the percentage change an 80% power at 5% sig level, my study will be able to detect. I am using the pwr package in R and the following formula-


With this I get an effect size of 0.034.

Is there a way, I can calculate as to how much of a percentage change will be detected using 80% power and 5% sig level between n1 and n2?

Is unary language with polynomial power context sensitive?

I suppose that $Sigma = {a}$.

Prove or Disprove: For every polynomial $p(n)$ with coefficients in $mathbb{N}$, $L = {a^{p(n)} ; | ; n in mathbb{N}}$ is a context sensitive language.

It seems that it is a context sensitive language. I guess making LBA or context sensitive grammar is not easy for this language. Can I prove this with closure property of CSL for example like complement? Can any one help me to prove for instance $L_1 = {a^{n^7+n^5+n^3+n^2+1} | nin mathbb{N}}$ is context sensitive. Maybe I can get an idea from this to prove my first question.