1B Statistics Flashcards

Question

What is the Poisson distribution and its uses?

Answer 1

Poisson distribution shows the frequency of events over time in which events occur independently. λ is the mean number of events per interval. This is equal to the variance. Example: deaths, hospital admissions X axis shows the number of events (x). Y axis shows the probability of x number of events. For small samples Poisson will be asymmetrical. For large it will approximate normal. Uses: - Used for discrete quantitative data e.g. counts or rates - Since it predicts randomly occurring events, it can be used to determine whether observed events are occurring randomly or not

Answer 2

Continuous probability distribution with a similar shape to the Normal distribution but with wider tails. Uses: - Small sample sizes e.g. n<30

Answer 3

The chi-squared distribution is continuous probability distribution whose shape is defined by the number of degrees of freedom. It is a right-skew distribution, but as the number of degrees of freedom increases it approximates the Normal distribution. Uses: - Used in chi-squared test

Answer 4

This is a false positive. It is when the null hypothesis is wrongly rejected i.e. the study shows an effect which in reality does not exist. Denoted by α. It is equivalent to the p-value e.g. 0.05 or 0.001. This is because the definition of the p-value is the probability that the result, or a more extreme result is due to chance. Alpha represents the same thing i.e. that the result is due to chance, not the true effect.

Answer 5

This is a false negative. It is when the null hypothesis is NOT rejected when it SHOULD HAVE BEEN. The study does not detect a difference that in reality existed. Occurs when sample size too small. Denoted by β

Answer 6

Power = 1 - β

Answer 7

1. Precision/Significance level aka p-value aka Type I error 2. Power (probability that study will be able to detect a true difference, where one exists 3. Clinically meaningful effect size 4. Prevalence. For cohort and intervention studies this is the disease prevalence in the unexposed. For case-control studies it is the prevalence of exposure in controls. But think logically about whether it is relevant to the study design e.g. won't be relevant to a smoking cessation intervention, where everyone already smokes. - Smaller significance level needs higher sample size - Higher power needs higher sample size - Smaller effect size requires larger sample size - Low prevalence requires large sample size

Answer 8

Running lots of tests means you will eventually get a clinically significant test even when the difference does not truly exist. For p=0.05 you would imagine 1/20 results would be a false positive. Can then lead to publication bias. Happens when: 1. Many outcomes are tested for significance 2. In a trial, one outcome is tested a number of times during the follow up 3. Many similar studies are being carried out at the same time. Ways to combat: 1. To specify clearly in the protocol which are the primary outcomes (few in number) and which are the secondary outcomes. 2. To specify at which time interim analyses are being carried out, and to allow for multiple testing. 3. To do a careful review of all published and also unpublished studies. Of course, the latter, by definition, are harder to find.

Answer 9

If doing n independent tests one should specify the type I error rate as α/n rather than α.

Answer 10

Simple data visualisation that displays frequency or distribution by placing dots (representing individual data points) above a labeled number line

Answer 11

Graph of continuous variable, grouped into several non-overlapping and equal intervals. The individual rectangles are called 'bins'. Usually would have 5-15 bins.

Answer 12

Shows the median, LQ, UQ and range. A variation is for the two ends of the whiskers not to be the range, and to have the outliers represented by dots. Useful for: - Showing skew via position of median relative to LQ and UQ - Comparing groups

Answer 13

Showing relationship between two continuous variables Advantages: - Retain all the data values - Make outliers apparent Disadvantages: - Hard to visualise individual results for very large datasets - Weak relationships may not be apparent

Answer 14

Categorical data. Note there is a gap between each bar. Height of the bar shows frequencies or relative frequencies.

Answer 15

Statistical technique used to measure the strength of linear association between two continuous variables. The correlation coefficient (r) lies between -1 and +1 (inclusive). If r = 1 or -1, there is perfect positive (1) or negative (-1) linear relationship If r = 0, there is no linear relationship between the two variables Conventionally 0.8 ≤ |r| ≤ 1.0 = very strong relationship 0.6 ≤ |r| < 0.8 = strong relationship 0.4 ≤ |r| < 0.6 = moderate relationship 0.2 ≤ |r| < 0.4 = weak relationship 0.0 ≤ |r| < 0.2 = very weak relationship NB correlation only measures LINEAR relationships. A U-shaped relationship may have a correlation of 0.

Answer 16

When calculated using the observed data, it is commonly known as Pearson's correlation coefficient. When using the ranks of the data, instead of the observed data, it is known as Spearman's rank correlation.

Answer 17

The square of the correlation coefficient (r2) indicates how much of the variation in variable y is accounted for (or “explained”) by the variable x. For example, if r = 0.7, then r2 = 0.49, which suggests that 49% of the variation in y is explained by x.

Answer 18

y = a + bx a = constant (y intercept) b = gradient (regression coefficient) The model is fitted by choosing a and b such that the sum of the squares of the prediction errors (the difference between the observed y values and the values predicted by the regression equation) is minimised. This is known as the method of least squares. The method produces an estimate for b, together with a standard error and confidence interval. From this, one can test the statistical significance of b. In this case, the null hypothesis is that b = 0, i.e. that the variation in y is not predicted by x.

Answer 19

Can be used for quantitative response variables with either continuous or categorical explanatory variables.

Answer 20

Used when the response variable is binary, being either an event (e.g. death or cure) or no event (e.g. survival or not cured). The explanatory variables can be either binary, ordinal, categorical or continuous.

Answer 21

1. Cohort life tables These show the probability of death at each age in a described group of individuals that has been followed over time. Cohort life tables are frequently used for survival analyses. This tracks a specific birth year across their life, including projected future mortality improvement. 2. Period life tables These give the current probability of death in given population at different ages. Period life tables are often used in demography. This uses current mortality rates only.

Answer 22

- Any curve that shows probability of surviving beyond a given time - Y-axis: survival probability (0–1) - X-axis: time - Can be theoretical, model-based, or empirical

Answer 23

- Time to event outcome - Follow up times differ - Censoring present

Answer 24

- Time to death - Time to relapse - Time to first event

Answer 25

A censored observation is one where there is incomplete data about when exactly the person experienced the outcome. Censored observations occur in two main ways: 1. Before the study completes, a subject may withdraw, or be lost to follow-up. 2. On completion of the study, subjects who have not yet experienced an event.

Answer 26

Hazard = instantaneous event rate at any given time (not a risk, not a probability) Hazard ratio compares hazards between groups HR < 1 → lower event rate in exposed group A hazard ratio of 0.75 suggests a 25% lower event rate at any given time in the intervention group. NB hazard ratio is NOT equivalent to the risk ratio

Answer 27

- A non-parametric estimate of the survival function - Built directly from observed event times - Accounts explicitly for censoring - Stepwise drops occur only at event times NB there is no adjustment for covariates, KM is not regression it is JUST descriptive

Answer 28

1. Starting point - All groups start at survival = 1.0 2. Drops in the curve - Each drop = an event - Bigger drops = more events at that time 3. Censoring - Tick marks show censored observations - Censored individuals contribute follow-up time up to that point - The censored individuals are those whose follow-up ended without an event e.g. loss to follow up - Therefore KM curve does not drop because they was no event 4. Separation of curves - Persistent separation suggests a difference in survival experience 5. Number at risk - Reliability decreases as numbers at risk fall - Late divergence should be interpreted cautiously NB they do NOT show hazards or hazard ratios

Answer 29

- Null hypothesis: no difference in survival functions - Compares observed vs expected events over time - Gives a global p-value NB does not give an effect size, does not adjust for covariates, no info on how big the difference is

Answer 30

What proportional hazards means - The ratio of hazards between groups is constant over time. - In plain English: One group is consistently “riskier” than the other The relative difference does not change over time When proportional hazards is plausible: - Curves separate early - Remain roughly parallel - Do not cross When proportional hazards is violated: - Curves cross - Separation changes markedly over time - Early benefit disappears or reverses Exam sentence - The roughly parallel separation of the curves suggests that the proportional hazards assumption is reasonable. Or, if violated: - The crossing of the curves suggests that the proportional hazards assumption may not hold.

Answer 31

- Semi-parametric statistical method - Used to in survival analysis i.e. for time to event data - Estimates a hazard ratio comparing groups - Allows adjustment for multiple covariates, including changes in co-variates over time - Allows for censoring - Does not involve specifying the baseline hazard

Answer 32

What the hazard ratio from Cox means: - The hazard ratio represents the relative event rate at any given time, averaged over the follow-up period, assuming proportional hazards. Example interpretation: - A hazard ratio of 0.70 indicates a 30% lower event rate at any given time in the exposed group, assuming proportional hazards.

Answer 33

If proportional hazards are violated, the Cox model still provides a summary estimate, but the hazard ratio should be interpreted as an average effect over time.

Answer 34

KM: - Descriptive - No adjustment for covariates - Stepwise - Non parametric - Difference in survival tested with a log rank test Cox adjusted curves - Model based - Adjust for covariates - Often smooth - Semi-parametric - Can calculate a hazard ratio (effect estimate) and CI and p-value

Answer 35

Heterogeneity means that the results of the included studies are genuinely different from each other beyond what would be expected by random sampling error alone. Put simply, the effect sizes are not all estimating the same underlying “true” effect. Therefore it may not be appropriate to pool them.

Answer 36

Clinical heterogeneity refers to differences in the specific research question that was studied, such as differences in the eligible populations, in the interventions and controls, and in the outcome measures. Methodological heterogeneity describes a variability in study design and in the risk of bias. This can include differences in the interventions given, and in how the outcomes were defined and measured, as well as variations in the use of blinding and allocation concealment. Such methodological heterogeneity may result in different studies actually measuring slightly different things. Statistical heterogeneity refers to variability in the “true” intervention effects in different studies, and it arises as a consequence of clinical and/or methodological heterogeneity. It results in a variation in effect sizes that are larger than can be expected by chance. Statistical heterogeneity can be identified using Cochran’s Q statistic (a form of chi-squared test of the null hypothesis that the true effect in all included studies are the same), or the I2 test (which uses Cochran’s Q statistic to give a percentage score for heterogeneity, with higher percentages indicating greater heterogeneity).

Answer 37

This is calculated as the weighted sum of squared differences between the effects from individual studies and the pooled effects from all included studies. The Q statistic has a chi-square distribution with (k-1) degrees of freedom, where k is the number of included studies. The resulting Q statistic can be used to generate a p value for the null hypothesis of no heterogeneity. Note that Cochran’s Q has a low power to detect heterogeneity when the number of studies is small (e.g. < 20), as is the case with most meta-analyses. To compensate for this, a higher significance level may be used to determine statistical significance (e.g. p < 0.10).

Answer 38

The I2 statistic estimates the proportion of variation across included studies that is secondary to heterogeneity (rather than chance). It is calculated using the Q statistic. An I2 of zero means that all the variability in effect sizes seen is due to sampling error and not heterogeneity. An I2 value of above 30% may represent at least moderate heterogeneity, but this result needs to be interpreted in context of the actual clinical or methodological features that may have led to the heterogeneity.

Answer 39

A funnel plot is a specific type of scatterplot, plotting the intervention effect sizes from different studies (on the x-axis) against some measure of the study size or precision (e.g. the inverse of standard error, on the y-axis).

Answer 40

Because the precision of the estimate of the effect size increases with the size of the study, the smaller studies will have more widely scattered effect sizes towards the bottom of the scatterplot, and this variability will reduce as the study sizes increase. The premise is that publication bias will result in smaller studies with non-significant outcomes not being published. If publication bias is present it will result in an asymmetric appearance of the funnel plot, with a unilateral gap towards the bottom of the funnel where the results of the small, negative, unpublished studies should have been. Techniques exist to modify summary estimates based on funnel plots using statistical estimation of missing data e.g. “trim and fill”.

Answer 41

It will result in an overestimation of the true treatment effect.

Answer 42

More than 3.84 Chi squared of less than 3.84 is therefore not significant at the 5% level

Answer 43

McNemar's test - This is specifically for paired data e.g. case-control study or repeated measurements in one participant - Data is presented in matched pairs - each cell is a matched pair - Therefore the total of the values is half the sample size - Interested in DISCORDANT PAIRS - Case exposed/control unexposed = r - Case unexposed/control exposed = s OR = r/s McNemar's = (r-s)SQUARED/(r+s)

Answer 44

Absolute risk reduction = risk in A - risk in B Relative risk reduction = (risk in A - risk in B)/risk in A

Answer 45

- Used for comparing the means of an outcome variable across two or more exposures variables, by comparing within-group and between-group variance - It is a special case of multiple regression - Most datasets for which ANOVA is appropriate can be analysed by regression too and yield the same results ONE-WAY ANOVA - One way analysis of variance is used when the exposure groups being compared are defined by one exposure e.g. socioeconomic status --> Assesses how much of the overall variation in the outcome is attributable to differences between the exposure group means --> Compared using an F-test, sometimes called the variance-ratio test --> Where there are only two groups, the one-way ANOVA gives exactly the same result as a t-test TWO WAY ANOVA - Two-way analysis of variance is used when subdivision is based on two factors e.g. age and sex MANOVA (Multivariate analysis of variance) - Comparing two outcome variables simultaneously across exposure sub-categories - E.g. the means of dependent variances reading, writing and maths may be tested across due exposure groups male and female

Answer 46

Prob (A) = Odds (A)/ (1 + Odds (A)) Odds (A) = Prob (A)/(1 - Prob (A)) Therefore when probability is small, odds and probability will be similar. This is because in this situation (1-Prob A) is close to 1, so you are literally just dividing by close to 1, so you will get a similar number Odds are always bigger than probability, since (1 - Prob) is less than 1 (so you are dividing by a number less than one)

Answer 47

- Sampling distributions are normally distributed - Data are measured at the interval, or ratio level (i.e. they are continuous) - Homogeneity of variance of the populations - Scores are independent - No extreme outliers

Answer 48

- The differences in each pair are normally distributed - Data are measured at the interval, or ratio level (i.e. they are continuous) - Data consist of two categorical related groups e.g. same subject before and after - The pairs (not the individuals within each pair) are independent of each other - No extreme outliers

Answer 49

Continuous (normal) outcome: Paired t-test Categorical outcome: McNemar's Ordinal, or continuous non-normal: Wilcoxon

Answer 50

Paired t-test --> Wilcoxon Signed Rank Unpaired t-test --> Mann-Whitney U test Pearson --> Spearman ANOVA --> Kruskal-Wallis

Answer 51

Two-sided P values are a test of a non-directional hypothesis that are used when we don't know which way the exposure will influence the outcome (increase vs decrease).

Answer 52

It would mean that the differences in size of the practice lists (unequal homogeneity of variance) would give rise to unequal standard deviations for those populations (within groups) and lead to an invalid result. Logarithmic transformation of list sizes will reduce the within group error variance.

Answer 53

- Non-parametric equivalent of ANOVA - Rank based test - Used to compare whether two or more independent groups differ

1B Statistics Flashcards

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