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Flashcards

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Log-Rank Test

Used to compare survival distributions of two samples. Formula: The calculation involves event times and survival probabilities, but exact formula is complex and often done with specialized survival analysis software.

Pearson Correlation Coefficient

Measures the linear relationship between two continuous variables. Formula:

r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}

where $x_i$ and $y_i$ are the values of the two variables.

Confidence Interval for Mean

Estimates the interval in which the population mean is likely to lie with a certain level of confidence. Formula for large sample sizes ( $n>30$ ):

\bar{X} \pm Z_\alpha\frac{\sigma}{\sqrt{n}}

Formula for small sample sizes (

n \le 30

\bar{X} \pm t_\alpha\frac{s}{\sqrt{n}}

where $\bar{X}$ is the sample mean, $\sigma$ and $s$ are the population and sample standard deviations, respectively, and $n$ is the sample size.

Paired T-Test

Compares means of two related groups (e.g., before-after, matched pairs), used with small sample sizes. Formula:

t = \frac{\bar{d}}{s_d/\sqrt{n}}

where $\bar{d}$ is the mean of the differences and $s_d$ is the standard deviation of the differences.

Wilcoxon Signed-Rank Test

Compares two related samples to assess differences in their population mean ranks. Formula: The test statistic is based on the sum of signed ranks, but the formula is complex and typically done with software.

T-Test for Mean (One-sample)

Used when the sample size is small ( $n \le 30$ ), population variance is unknown, and the data is approximately normally distributed. Formula:

t = \frac{\bar{X} - \mu}{s/\sqrt{n}}

where $\bar{X}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size.

Chi-Square Test for Independence

Determines if there is a significant association between two categorical variables. Formula:

\chi^2 = \sum \frac{(O - E)^2}{E}

where $O$ represents the observed frequency and $E$ represents the expected frequency under the null hypothesis.

Mann-Whitney U Test

Compares differences between two independent groups when the assumptions for the t-test are not met (e.g., non-normal distributions). Formula: The calculation involves ranking all observations and computing U based on these ranks, but the exact formula is complex and often calculated with software.

Kruskal-Wallis Test

Non-parametric version of ANOVA, used for comparing three or more independent samples of different sizes. Formula:

H = \frac{12}{N(N+1)}\sum \frac{R_i^2}{n_i} - 3(N+1)

where $N$ is the total number of observations, $R_i$ is the sum of ranks for the ith group, and $n_i$ is the number of observations in the ith group.

T-Test for Independent Samples

Compares means of two independent groups, used when variances are unknown but assumed to be equal, sample sizes may be unequal. Formula:

t = \frac{\bar{X}_1 - \bar{X}_2}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}

where $s_p$ is the pooled standard deviation.

One-Way ANOVA

Tests if there are any statistically significant differences between the means of three or more independent (unrelated) groups. Formula:

F = \frac{MS_{between}}{MS_{within}}

where $MS_{between}$ and $MS_{within}$ are the mean squares between and within groups.

Z-Test for Mean

Used when sample size is large ( $n>30$ ), population variance is known, and data is normally distributed. Formula:

z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}

where $\bar{X}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Two-Way ANOVA

Tests the effect of two independent categorical variables on a continuous outcome variable. Formula: It involves computing F-statistics for each variable and their interaction, but the formula is complex and beyond the scope of a flashcard.

Simple Linear Regression

Evaluates the linear relationship between two continuous variables. Formula:

y = \beta_0 + \beta_1x

where $y$ is the dependent variable, $x$ is the independent variable, $\beta_0$ is the intercept, and $\beta_1$ is the slope.

Spearman's Rank Correlation Coefficient

Measures the strength and direction of the association between two ranked variables. Formula:

r_s = 1 - \frac{6\sum d_i^2}{n(n^2 - 1)}

where $d_i$ is the difference between the ranks of each observation, and $n$ is the number of observations.

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