# Post-test Analysis

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There are three separate ways of estimating how treatments will work for an initial test. All three of these products produce mathematics equivalent results indicating that they gave each the same answer. We'll calculate the standard error of the difference between the two means in this context. This common error introduces information about the variance (standard deviation) of one or more of the two categories. The results from the three methods are just like each other. You will find there are statistical models and techniques used for all experiments and quasiexpérimental tests in regression constrained repeated measure model terms.

## Are you analyzing pre-post data with repeated measures ANOVA or ANC?

Pre and post-test scores can be analyzed with either repeated measure ANCOVA or Percent More Likely (a term used to analyze quasiexpérimental data). However, when analyzing pre-post tests such as the one you found, it works best to use percent more likely.

What is a p-value? And how does it relate to the probability that my results are accurate?

P-values tell you how likely it is that your result happened by chance.

It's likely, but not sure that people who saw one advertisement were more likely to buy than those who saw the other ad. This does not mean simply because one group was slightly more likely to buy than the other that the advertisement was successful. It is possible, for example, that people in one group tended to buy anyway, so any difference between the treatment and control groups would be an artifact of history.

The p-value lets you assess whether your results are accurate or due to chance. If a result has a small p-value, it might be real. The smaller the p-value, the more specific you are that it's real.

The formula for calculating a p-value is:

p = probability of getting results at least as extreme as yours if they were due to chance alone. So, in our example above, we know that 25% of people who saw the ad with testimonials bought within a week compared to 20% of people who saw the other ad. So there's only one chance in four that these results would have happened if both ads worked equally well.

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## How do I calculate bias?

Bias is the same as error, but it is used when estimating parameters rather than testing hypotheses. It is the difference between what you count and what is 'true, or at least closer to accurate than any other possible value.

In our example, let's assume that the actual conversion rate for those who saw one of two ads was 19%. Bias would measure how far from this parameter we estimated it to be, measured in terms of the standard error.

If we calculate this in our example, we get a bias of 1% towards assuming that both ads worked equally well. This might explain why an allocation was even. A trim level of causation can be hidden away in the statistical repeated measures analysis noise.

## How can I tell whether my results are significant?

Be careful when you see reports about 'statistically significant' differences between groups. It does not mean that a result is accurate or that one treatment has worked better than another. It only means that there was only a tiny chance of getting this result by chance.

It sounds like you're interested in more than just reporting significance. A better study would be one in which the probability has been established that one treatment is causally better than another, with a sufficiently small p-value.

## What is regression to the mean?

Regression to the mean occurs when people find that an extreme value of one variable tends to be followed by less extreme importance of the same variable. It is important because it seems like something magical happened when it was just due to chance.

This can happen when you pre-test people who are particularly good or bad at something and appear to get even better or worse over time. In practice, it can be challenging to study people who are at the extremes, and you should try to understand whether regression to the mean is a natural effect or an artifact of your data collection.

If we assume that there was no bias in our experimental design, what might have caused an increase?

In this example, it's possible that there was no actual effect on pre-test scores. It could have been that something else changed in the market at the same time as your advertising campaign, or some people were simply more likely to buy during this period.

## What is the 'rule of three' for sample size?

The rule of three states that you should aim for 30 subjects per treatment group if you are making a comparison that will require complex statistical analysis (such as multiple regression) and about ten if you are making simple comparisons.

However, this may be unsafe in some cases because it does not take into account the effect size expected. For example, suppose your study is designed to compare two groups with an observational study. In that case, you can assume that any statistically significant result is an effect that you cannot ignore.

## What are the problems with not measuring how significant the effects are?

One of the most significant issues in experimental design is missing out on a discussion of power. If you do not consider the chances of finding a difference, then it can seem like one treatment has worked better than another when this isn't the case. Therefore, it is best to go into a study with an idea of how large an effect you expect.

## How can I compare my experimental results against a control group?

If you have a control group, you must take care not to include it in any analyses, which will bias your findings. You may need to repeat data collection on them and your experimental group, for example, to measure the effect of time passing.

This means that you cannot compare results between groups with complex statistical analyses such as regression. But, of course, this is not a problem if you want to understand whether one treatment worked better than another and don't care how big this difference was.

## Would it be appropriate to present the results with a p-value?

Yes, but you must also present confidence intervals with your post-test score. A 95% CI will allow an audience to appreciate the size of the effect and whether it is meaningful for them. These intervals should be narrow if you have collected your data well.

## To measure real effects by finding a small p-value, why do we need to collect data from a larger sample size?

The smaller the probability is that an effect occurred by chance, the more likely it is that you have found an actual impact. In general, as your experiment becomes more extensive, you will be able to pick up on more minor effects with a given level of confidence. This is why you need to collect more data from a larger sample size.

## What is the purpose of a power analysis?

A power calculation estimates how likely your experiment will find some effect. For example, it tells you how large your experimental group needs to be for there to be some chance of finding a difference between groups if the truth is there.

## How can I calculate the power of an experiment?

The calculation for power is 1-rel/k, where k is the number of groups you are comparing. So if your new treatment has no effect, it will be pretty easy to find this out by having a large sample size because you're not trying to measure anything small.

## What are the limitations of post-hoc power analyses?

It would be best if you did not try to predict the outcome of your experiment before you conducted it. You cannot trust this number; it is based on many assumptions about the experimental design, which you may find false in practice. It also only applies if your groups are balanced so that no biases creep in that work against a treatment.

## What is the purpose of a sample size calculation?

A sample size calculation tells you how many data points you need to measure in your experiment. It considers the effect size and compares this to the margin of error from not being on target with your experimental design.

It is essential to have enough participants in your experiment that you avoid a significant margin of error. If this is too large, you will end up with a small p-value while the effect size is not substantial enough to be meaningful.

## How can I calculate how many participants I need?

You can use an online calculator like nQuery to work out this for yourself. The process requires you to estimate your effect size and how much margin of error you think is acceptable. These estimates will be based on what kind of experimental design and expected variation in the results is possible for your research.

## What is the relationship between sample size, power, and effect size?

Effect size is one-half of the ratio between standard deviation (SD) and the standard error of the mean (SEM). For example, if your SD is equal to 5, you have a 25% effect size. You can tell how precise your experiment needs to be based on this.

You will require a smaller effect size to see differences between groups with large sample size. This is why it is essential to have enough participants.

## How can sample size be reduced?

You can reduce your sample size by using the lowest possible standard deviation, which will allow you to have a more significant effHowever, the size. The ratio between SD and SEM remains unchanged so that you keep the precision of your experiment at the same level. Because of this, there is an exponential relationship between the effect size and SEM.

## What is a confidence interval?

A confidence interval is a range of values likely to contain the actual population value. To make this more comprehensive, you need to increase your margin of error, which increases the number of data points you collect. This means that every time you reduce your margin of error by collecting more data, you will increase your number of data points at the same ratio.

The width of the confidence interval is inversely proportional to the square root of n, where n is the number of participants. This means that the margin of error halves for every new participant you add, and so does your confidence interval.

## What makes up the confidence interval?

The margin of error is not the same as the width of your CI. The width of your CI is based on two things: the SEM and how confident you need to be that your interval contains the actual population value. You will always have a 5% chance that you are wrong, so this means that if you want to be 90% confident of your CI containing the actual population value, you set SEM equal to 2.5 times the SD of your group.

For example, if you have a 30% effect size with an SD of 5, this works out at 1.5 * 5 = 7.5. This is the amount by which the width of your CI is above and below (or before and after) the value of your experimental group.

## Can you have a confidence interval without doing any analysis?

Yes, because all the information about the confidence intervals is already available in your sample size calculation output. You can find these values directly within the display at your p-value>0.05 (for both groups). You can take the SEM value from here to see how broad your CI will be.

## How do I calculate power?

Power is used to tell you if you needed more participants (and hence a bigger sample size) or not, based on how much variation you expected in your experiment. Power can be calculated by using the formula power.

If you expect more variation in your experiment, you will need a higher power and vice versa. This is because sample size changes as a ratio to expected variation. A common mistake with this is using spreadsheet software like Excel, where it won't be possible to predict how much variation you will get in your experiment.

## How do I calculate the sample size?

There are three different ways to do this, depending on how much information you have available about the population of interest. The first way is to use a power calculation calculator, which allows you to specify what effect size and power you want to achieve with your experiment. The second way is to use a nomograph, which will enable you to tell your figures from previous experiments and a three-figure lookup table.

The third way to calculate the sample size is using an online calculator. Examples of these can be found here or here. This is the quickest and easiest option, but there are some caveats. The main problem with these calculators is that they rely on a normal distribution, which depends on having enough data to get a good idea of how much variation you will get in the following experiment.

## If I collect more data, what does this mean for my power?

It means that your power will increase because it's like moving to the right on the graph. However, if you collect more data for the same sample size, your energy will increase slightly. The reason is that with a larger sample size, most of the variance in your results comes from random error instead of systematic error (which lowers your p-value and confidence intervals).

## If I collect more data, what other effects will there be?

There are two main things to look out for: an increase in your confidence interval and a change in the conversion rate. Your CI should get wider because you have more data. If this is a problem, you need to do a power calculation before collecting any additional data. The effect on your treatment's conversion rate is more complicated to predict because this depends on the statistical power of your experiment. If you're using an online calculator, it may be best to increase your sample size slightly (between 5-10%) to account for the variance in both groups.

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