Introduction to Biostatistics

1 Learning objectives

1.1 Learning objectives

  • Compare and contrast point estimates and interval estimates
  • Construct confidence intervals
  • Interpret confidence intervals

2 Review

2.1 Sample vs population

flowchart LR
  A(Population) --> B{Sampling}
  B{Sampling} --> D[Sample]

  • Sample from a population

flowchart RL
  C{Inference} --> A(Population)
  D[Sample] --> C{Inference}

  • Make inference from sample back to population

2.2 Sampling distribution

  • Probability distribution
  • Quantifies the uncertainty in sampling from a population
    • Sample isn’t representative of the whole population
    • Sampling variability in the test statistic
    • Across all samples from this population, how variable can we expect the test statistic to be?
  • Standard error is the standard deviation of the sampling distribution

2.3 Sampling distribution

  • Sampling distribution depends on
    • Population distribution (sometimes)
    • Sample size
    • Test statistic (e.g., mean, regression coefficient)
  • In last week’s lab, we demonstrated the central limit theorem

2.4 Last week

  • Different population distributions
    • Normal, Bernoulli, binomial, Poisson
  • Sample 10000 times from each population
    • With different sample sizes: \(n = 10, 50, 100\)
  • Calculate sample mean (\(\bar{X}\)) for each sample
    • Examine distributions of means

2.5 \(population \sim \mathcal{N}(0, 1)\)

  • \(n = 10\)

  • \(n = 50\)

  • \(n = 100\)

2.6 \(population \sim \mathcal{N}(0, 1)\)

\(n\) \(\mu\) mean(\(\bar{X}\)) \(\sigma^2\) var(\(\bar{X}\)) \(\frac{\sigma^2}{n}\)
10 0 -0.003408 1 0.0980502 0.1
50 0 -0.0009473 1 0.0200694 0.02
100 0 0.0011459 1 0.010093 0.01

2.7 \(population \sim B(0.3)\)

  • \(n = 10\)

  • \(n = 50\)

  • \(n = 100\)

2.8 \(population \sim B(0.3)\)

\(n\) \(\mu\) mean(\(\bar{X}\)) \(\sigma^2\) var(\(\bar{X}\)) \(\frac{\sigma^2}{n}\)
10 0.3 0.29996 0.21 0.0216842 0.021
50 0.3 0.300286 0.21 0.0041745 0.0042
100 0.3 0.301151 0.21 0.0021253 0.0021

2.9 Central limit theorem

  • Regardless of the population distribution
    • the sampling distribution of the mean
    • follows a normal distribution
      • \(\bar{X} \sim \mathcal{N}(mean=\mu, variance=\frac{\sigma^2}{n})\)
    • as \(n \rightarrow \infty\)
      • Can happens very quickly: often \(n \ge 30\)

3 Point estimates

3.1 Point estimates

  • Single number estimate of the population value
    • Sample mean
    • Sample variance
    • Mean difference
    • Correlation coefficient

3.2 Estimation methods: Point estimates

  • Sample estimate
  • Maximum likelihood
  • (Ordinary) least squares: “OLS regression”
  • (Weighted) least squares

3.3 Maximum likelihood: Conceptual

  • Likelihood function based on the distributions of the variable(s)
    • Likelihood is similar to probability
      • Doesn’t have some properties
    • Likelihood of different potential parameter values (e.g., means)
    • Maximum likelihood estimates are the parameter estimates that are most likely given the data

3.4 Maximum likeihood: Figure

  • Maximum likelihood estimate is found at \(\mu = 15.4\)

Figure 2 from Enders, C. K. (2005). Maximum likelihood estimation. Encyclopedia of statistics in behavioral science.

4 Interval estimates

4.1 Estimation methods: Interval estimates

  • Known formulas for some situations
  • Monte Carlo simulation
  • Resampling methods: bootstrap, jackknife (leave one out), etc.
  • Other computational methods

4.2 Why an interval?

  • Point estimate is almost certainly wrong
    • e.g., single mean from a single sample
  • Point estimates are random variables
    • Random variables have a distribution
    • Distributions tell us about probability
    • What are the most likely values?

4.3 Normal distribution

  • Remember: Sampling distribution is normal
    • Sample means follow a normal distribution
    • If we want to talk about this sample mean in the context of all sample means, then we can use the sampling distribution

4.4 Aside: \(z\)-scores

  • \(z\)-scores are scores on a standard normal distribution
    • \(\mathcal{N}(0,1)\)
  • Mean = 0
  • Variance = standard deviation = 1
  • Often called “standardized scores”: Subtract mean, divide by SD

4.5 (Standard) Normal: mean = 0, SD = 1

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  xlim(-4, 4)

4.6 Symmetric around the mean

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(0, 4)) +
  annotate("text", x = 0.8, y = 0.1, label = "50%", color = "white", size = 6) +
  annotate("text", x = -0.8, y = 0.1, label = "50%", size = 6) +
  xlim(-4, 4)

4.7 Proportion for each standard deviation

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-3, -2)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-1, 0)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(1, 2)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(3, 4)) +
  annotate("text", x = -3.5, y = 0.05, label = "0.13%") +
  annotate("text", x = -2.5, y = 0.1, label = "2.14%") +
  annotate("text", x = -1.5, y = 0.05, label = "13.59%") +
  annotate("text", x = -0.5, y = 0.2, label = "34.13%", color = "white") +
  annotate("text", x = 0.5, y = 0.2, label = "34.13%") +
  annotate("text", x = 1.5, y = 0.05, label = "13.59%", color = "white") +
  annotate("text", x = 2.5, y = 0.1, label = "2.14%") +
  annotate("text", x = 3.5, y = 0.05, label = "0.13%") +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.8 Between -1 SD and +1 SD: 68.26%

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-1, 1)) +
  annotate("text", x = 0, y = 0.1, label = "68.26%", color = "white", size = 6) +
  annotate("text", x = 2.5, y = 0.1, label = "15.87%", size = 6) +
  annotate("text", x = -2.5, y = 0.1, label = "15.87%", size = 6) +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.9 90% between -1.645 SD and +1.645 SD

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-1.645, 1.645)) +
  annotate("text", x = 0, y = 0.1, label = "90%", color = "white", size = 6) +
  annotate("text", x = 3, y = 0.05, label = "5%", size = 6) +
  annotate("text", x = -3, y = 0.05, label = "5%", size = 6) +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.10 95% between -1.96 SD and +1.96 SD

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-1.96, 1.96)) +
  annotate("text", x = 0, y = 0.1, label = "95%", color = "white", size = 6) +
  annotate("text", x = 3, y = 0.05, label = "2.5%", size = 6) +
  annotate("text", x = -3, y = 0.05, label = "2.5%", size = 6) +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.11 99% between -2.326 SD and +2.326 SD

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-2.326, 2.326)) +
  annotate("text", x = 0, y = 0.1, label = "99%", color = "white", size = 6) +
  annotate("text", x = 3, y = 0.05, label = "0.5%", size = 6) +
  annotate("text", x = -3, y = 0.05, label = "0.5%", size = 6) +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.12 Confidence intervals

  • Interval estimate of likely values for an estimate
    • Sample mean is 52.1, but the population mean is unlikely to be exactly that
    • Range of values in which the population mean exists
  • Leverage normal distribution of sampling distribution
    • Use proportion of distribution between different values

4.13 \(100 \times (1-\alpha)\%\) CI for mean

  • \([\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]\)
    • \(z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}\): margin of error
    • \(\bar{X}\): sample mean
    • \(\alpha\): probability of type I error (probability in the tails)
    • \(z_{1-\alpha/2}\): \(z\)-score corresponding to a specific \(\alpha\)
    • Example: \(95\%\) CI: \(\alpha\) = .05, \(z_{1-\alpha/2}\) = 1.96

4.14 Percentiles

Code 
ggplot(data.frame(x = c(-4, 4)), aes(x)) +
  stat_function(fun = dnorm,
                geom = "line",
                xlim = c(-4, 4)) +
  stat_function(fun = dnorm,
                geom = "area",
                fill = "steelblue",
                xlim = c(-1.96, 1.96)) +
  annotate("text", x = 0, y = 0.1, label = "95%", color = "white", size = 6) +
  annotate("text", x = 3, y = 0.05, label = "2.5%", size = 6) +
  annotate("text", x = -3, y = 0.05, label = "2.5%", size = 6) +
  geom_vline(xintercept = -3, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = -1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 1, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 2, linetype = "dashed", alpha = 0.3) +
  geom_vline(xintercept = 3, linetype = "dashed", alpha = 0.3) +
  xlim(-4, 4)

4.15 Percentiles from qnorm() function

  • Percentile: proportion of distribution to the left of a score
  • For a 95% CI, you do not want the 95th percentile
    • 2.5% in the left tail, 95% in middle, 2.5% in right tail
    • 97.5th percentile (95 + 2.5 = 97.5) provides the \(z\)-score you want
qnorm(0.975)
[1] 1.959964
  • Default is mean = 0, SD = 1 but you can specify different in arguments
    • e.g., qnorm(0.975, mean = 5, sd = 2)

4.16 CI interpretation

  • One experiment: 95% chance the CI captures the population mean
    • Don’t say: “95% chance that the population mean is in the CI”
    • Population value is fixed; CI either captures it or not
  • Long run: Over repeated versions of the same experiment, \(100 \times (1 - \alpha)\%\) of the intervals will contain the population value
    • Population value is fixed; CI either captures it or not
  • Frequentist (vs Bayesian)

4.17 CI width

  • Decreases as \(n\) increases
    • Because the sampling distribution narrows as \(n\) increases
    • \(\frac{\sigma^2}{n}\) decreases as \(n\) increases
  • Increases as population variability increases
    • Because \(\frac{\sigma^2}{n}\) increases as \(\sigma^2\) increases
  • Increases as \(\alpha\) decreases
    • Because the \(z\)-score used increases as \(\alpha\) decreases

4.18 Example: \(95\%\) confidence interval

n <- 50
n
[1] 50
x <- rnorm(n, mean = 5, sd = 2)
datx <- data.frame(x)
meanx <- mean(x)
meanx
[1] 5.405786
sdx <- sd(x)
sdx
[1] 1.998746
varx <- var(x)
varx
[1] 3.994985

\([\bar{X} - z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}, \bar{X} + z_{1-\alpha/2}\ \sqrt{\frac{\sigma^2}{n}}]\)

\([5.406 - 1.96 \times \sqrt{\frac{3.995}{50}}, 5.406 + 1.96 \times \sqrt{\frac{3.995}{50}}]\)

\([5.406 - 0.554, 5.406 + 0.554]\)

\([4.85, 5.96]\)

4.19 What does it look like?

Code 
ci95min <- meanx - 1.96*sqrt(varx/n)
ci95max <- meanx + 1.96*sqrt(varx/n)
ggplot(data = datx, aes(x = x)) +
  geom_histogram(bins = 30) +
  geom_vline(xintercept = meanx) +
  geom_vline(xintercept = ci95min, linetype = "dashed") +
  geom_vline(xintercept = ci95max, linetype = "dashed") +
  scale_x_continuous(breaks = c(0:10))

4.20 Why is it so narrow?

  • Shouldn’t it cover 95% of the observations?
    • NO!!!
  • Remember: We don’t care about the sample
    • We care about the population
    • The confidence interval tells us about the population mean
    • Relatively narrow band is where we think the population mean is

5 Test hypotheses using CIs

5.1 Hypothesized population mean

  • Does sample come from population with a mean of [some value]?
    • Check whether the confidence interval contains [some value]
    • Yes: It could come from that population
    • No: We are [some %] confident it doesn’t
  • Example data: Does it come from a population with mean = 3?
    • 95% confidence interval is \([4.85, 5.96]\)
    • 3 is not in CI: 95% confident that this sample did not come from a population with mean = 3

5.2 Compare groups using CIs

  • Can we compare two groups using their confidence intervals?
    • Yes, but it’s not intuitive
  • If CIs don’t overlap
    • Groups are different
  • If CIs do overlap
    • The groups may or may not be different
  • Check References on course website for “Graphical comparisons”

6 In-class activities

6.1 In-class activities

  • Construct some confidence intervals
  • Interpret the confidence intervals in words

6.2 Next two weeks

  • Hypothesis testing
    • One mean vs hypothesized value
    • Two means compared to each other