# 20.109(F21):M1D7

## 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).

Because standard deviation is only justified when sufficient data have been collected to generate a normal curve, you will use confidence intervals to report the likelihood that your results predict the true mean. A confidence interval is a defined interval that is calculated to define the true mean to a specified level of confidence. Simply, it is possible to define a range in your data set that likely contains the true mean based on the calculated mean.

• Confidence interval (CI) is defined as:

CI = $\overline{\chi } \pm \frac{ts}{\sqrt{n}}$, where t = value from t table (dependent on specified confidence level and n)

In your data, you should use the CI to generate error bars due the low n. Be sure to report which confidence level was used to calculate the intervals reported. 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 tcalculated. This tcalculated value is compared to the ttabulated 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 tcalculated value is greater than the ttabulated, 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 tcalculated for a data set with n - 1 = 10 is 3 (given that the ttabulated 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

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:

• Include a bar graph of the data with 95% confidence intervals.
• 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. confidence intervals) and significance (i.e. p-values).

For the γH2AX data:

In the analysis that you completed, you averaged the data from eight images for each condition. In addition, you used two different methods to analyze the raw data. For the figure that you will include in the Data summary, plot the averaged values then perform the statistical analysis on values calculated for the averaged datasets.

For the CometChip data:

In the analysis that you completed, you averaged three technical replicate samples. For a more robust data set, you will now include the results from a second experiment that was completed according to the exact same protcol. These data will serve as biological replicates for the conditions tested (dataset #2 linked here).

The values provided in dataset #2 are presented in a layout similar to the data you analyzed M1D6 (dataset #1); however, in dataset #2 the triplicate samples are averaged for you. For the figure that you will include in the Data summary, average the values from the two datasets and plot using a line graph. Then perform the statistical analysis on the values calculated for the combined dataset.

In your laboratory notebook, complete the following:

• Review the values presented in each of the datasets.
• Are the data in agreement (ie do the different datasets look similar as far as the overall results / trends are concerned?)?
• Are there any discrepancies between the datasets?