# 20.109(F22):M1D7

## Contents

## Introduction

Today is the final laboratory session for Module 1! You have completed all of the bench work for your research; however, there is still data analysis to complete for your experiments. In addition to plotting the data, you will complete statistical analysis to determine the significance of your results.

Statistics are mathematical tools used to analyze, interpret, and organize data. The specific tools that you will use are confidence intervals (CI) and the Student's *t*-test. To begin, review the following definitions:

- Mean (or average) is defined as:

$ \overline{\chi } = \frac{\sum_{i}^{n}\chi _{i}}{n} $, *where* $ \chi _{i} $ = *individual value and n = number of samples*

- With infinite data, the mean ($ \overline{\chi } $) approaches the true mean (μ).
- Standard deviation measures the variation in the data and is defined as:

$ s = \sqrt{\frac{\sum_{i}^{n }(\chi _{_{i}}-\overline{\chi })}{n - 1}} $, *where n - 1 = degrees of freedom*

- With infinite data, the standard deviation (
*s*) approaches the true standard deviation (σ).

An assumption is made when using standard deviation to report the variation in a data set. It is assumed that sufficient data have been collected to generate a normal curve.

So, what does this all mean in regard to the data you will report? As an example, if the calculated $ \overline{\chi } $ of a data set equals 80 au there is a 95% chance the μ is between 50 au and 110 au, where au = arbitrary units. And how does this relate to *s*? If you know the μ, the σ represents a 68% confidence interval.

When interpreting data, the error bars are representative of the noise in the data or how different the data points are for each of the replicates. Replicates come in two types: technical and biological. Technical replicates indicate that the same sample was tested multiple times and is measure of experimenter error (for example, pipetting errors between aliquots). Biological replicates indicate that different preparations of the same sample were tested and is a measure of the difference in a response to a variable (for example, response to a treatment between separate cultures of the same cell line). Though both types have value in data analysis, the interpretation of the error represented in each case is different. Because of this it is important to indicate if the replicates used in the data analysis are technical or biological. For your data, what type of replicates did you analyze for the γH2AX experiment? For the CometChip experiment?

Lastly, you will use Student's *t*-test to report if your data are statistically different between treatments.

- Student's
*t*-test is defined as:

$ t = \frac{\left | \overline{\chi_{_{1}}} - \overline{\chi_{_{2}}} \right |}{s_{pooled}}\sqrt{\frac{n_{1} n_{2}}{n_{1}+n_{2}}} $, *where* $ s_{pooled} = \sqrt{\frac{s_{1}^{2} (n_{1} -1) + {s_{2}^{2} (n_{2} - 1)}{}}{n_{1} + n_{2} - 2}} $

The value you calculate with the Student's *t*-test equation is referred to as *t*_{calculated}. This *t*_{calculated} value is compared to the *t*_{tabulated} value in the the *t* table, according to the appropriate *n* - 1 using the p-value for the two-tailed distribution (which assumes that you do not know how the data will shift). If the *t*_{calculated} value is greater than the *t*_{tabulated}, then the data sets are significantly different at the specific p-value. So, what does this all mean in regard to the data you will report? As an example, if the *t*_{calculated} for a data set with *n* - 1 = 10 is 3 (given that the *t*_{tabulated} is 2.228), then the data sets are different with a *p*-value ≤ 0.05. Which means that there is less that a 5% chance that the data sets are the same.

## Protocols

### Part 1: Practice statistical analysis

If you would like additional practice in completing statistical analysis, please complete Part 1. If you are confident in your understanding, please proceed to Part 2.

Review data from an experiment where cells were exposed to increasing amounts of radiation (linked here). Your goal is to determine if a statistically significant amount of DNA damage was induced. For the purpose of this exercise, the values in the spreadsheet are in arbitrary units of 'DNA damage', where the higher numbers indicate more damage.

When interpreting the statistics, consider how you may use the information to convince someone that the DNA damage was significant. You may find the spreadsheet originally created by Prof. Bevin Engelward and modified for the 20.109 laboratory, helpful for this exercise (linked here).

**In your laboratory notebook,** complete the following:

- Attach the completed spreadsheet.
- Include a bar graph of the data with standard deviations.
- Indicate if there is a statistically significant difference (
*i.e.*provide a*p*-value) between the conditions tested.

### Part 2: Complete data analysis

Use the tools above to analyze the data for your γH2AX and CometChip experiments. The figures / analyses in your Data summary should include measures of variability (i.e. standard deviation) and significance (i.e. *p*-values).

**For the γH2AX data:**

In the analysis that you completed, you averaged the data from three images for each condition. Remember that you analyzed the data for your experiment and the data for a pilot experiment that was completed by the Instructors. Your data will be Panel A and the Instructor data will be Panel B of the final figure. For the figure that you will include in the Data summary, plot the averaged values for each experiment then perform statistical analysis on the calculated values for the averaged datasets.

**For the CometChip data:**

In the analysis that you completed for your experiment, you averaged the values of three technical replicates. For the figure that you will include in the Data summary, plot the averaged values for each condition then perform statistical analysis on the calculated values for the data set.

In the analysis that you completed for the timecourse experiment, you averaged the values of three technical replicates. For the figure that you will include in the Data summary, plot the averaged values according to the guidelines provided on M1D6 then perform the statistical analysis on the calculated values for the data set.

Next day: Complete in-silico cloning of protein expression plasmid