Applied Statistics 2 - University of Surrey PDF

VMS3012 Applied Statistics 2 Alice Batchelor Liz Grant Learning Outcomes Following on from Applied Statistics 1, we will look at some of the other statistical measures and analyses that you will encounter in research papers. Be able to calculate, interpret and Be able to use continge...

VMS3012 Applied Statistics 2 Alice Batchelor Liz Grant Learning Outcomes Following on from Applied Statistics 1, we will look at some of the other statistical measures and analyses that you will encounter in research papers. Be able to calculate, interpret and Be able to use contingency tables* explain the terms: Be able to interpret survival curves o risk reduction & NNT o absolute risk and relative risk o sensitivity and specificity o predictive values o likelihood ratios o odds, odds ratios o probabilities o pre- and post-test odds* * Some of this is a recap of VMS2008 statistics material UNIVERSITY OF SURREY 2 Types of research question 1 We will look at some of the statistical methods typically 4 used for these types of research questions and studies. 3 2 UNIVERSITY OF SURREY 3 Contingency tables LO: Be able to use contingency tables Recap: Contingency tables A contingency table is a table which displays the frequency distribution of two categorical (nominal or ordinal) variables One variable is shown in the rows, the other in the columns Most commonly, contingency tables are 2 x 2 (two rows, two columns) The table can be used to analyse the relationship between 2x2 contingency table the two categorical variables There are different methods for analysing contingency tables, depending on the nature of the study, data, research questions, goals of analysis o We saw in AS1 that we can use chi-square to test for association, but there are other measures which can give more meaningful insights in clinical contexts UNIVERSITY OF SURREY 5 Contingency tables: Common scenarios Therapy Diagnostic tests Aetiology Typical variables: Typical variables: Typical variables: Outcome (yes/no) True status (e.g. Outcome (yes/no) Treatment (yes/no or disease/no disease) Exposure (yes/no) experimental trt/usual trt) Test result (+ve/-ve) UNIVERSITY OF SURREY 6 Reading contingency tables To conduct and understand the various analysis methods associated with contingency tables, it is important to be confident in reading them, for example: Total number of animals Total number of cats who Number of animals exposed vaccinated? tested positive for FeLV? and with outcome present? 45 77 100 UNIVERSITY OF SURREY 7 Numeracy reminders Decimals vs. percentages We will frequently be switching between decimal and percentages: Multiply by 100 and add ‘%’ E.g. 0.45 = (0.45*100)% = 45% Decimal Percentage Remove ‘%’ and divide by 100 E.g. 75% = 75/100 = 0.75 Rounding Be careful with rounding values if these values will be used again for further calculations. UNIVERSITY OF SURREY 8  Statistics in research studies LO: Be able to calculate, interpret and explain the terms risk reduction & NNT, absolute risk and relative risk, sensitivity and specificity, predictive values, likelihood ratios, odds, odds ratios, probabilities, pre- and post-test odds LO: Be able to use contingency tables Treatment / Therapy LO: Be able to calculate, interpret and explain the terms risk reduction & NNT, absolute risk and relative risk, sensitivity and specificity, predictive values, likelihood ratios, odds, odds ratios, probabilities, pre- and post-test odds LO: Be able to use contingency tables Risk Risk is based on a proportion. It refers to the probability than an event will occur. Number of subjects/animals with event of interest: 𝑥 𝑥 Risk = Total number of subjects/animals at risk: 𝑛 𝑛 In studies, we often want to compare the risk of some outcome or event of interest between two groups e.g. Exposure groups (exposure vs. no exposure) – observational studies Treatment/therapy groups (treatment vs. placebo /non-experimental treatment) – ideally randomised control trials UNIVERSITY OF SURREY 11 Therapy: Example (Vaccination) This example will be used to demonstrate some of the analysis methods seen in this section. Therapy: Vaccination o Categories: vaccinated vs. unvaccinated Outcome (event): Disease o Categories: healthy (no event) vs. caught disease (event) UNIVERSITY OF SURREY 12 Control Event Rate (CER) and Experimental Event Rate (EER) The experimental event rate (EER) is the proportion of animals receiving the test treatment that have the event of interest. number of animals in treatment group with event EER = i.e. risk in treatment group total number of animals in treatment group The control event rate (CER) is the proportion of animals in the control group (placebo / no treatment / non-experimental treatment) that have the event of interest. number of animals in control group with event CER = i.e. risk in control group total number of animals in control group UNIVERSITY OF SURREY 13 CER and EER: Example (Vaccination) number of animals in treatment group with event EER = total number of animals in treatment group c 10 10 = = = = 0.1 = 𝟏𝟎% a+c 90 + 10 100 number of animals in control group with event CER = total number of animals in control group d 95 95 = = = = 0.95 = 𝟗𝟓% b+d 5 + 95 100 General note for calculations using contingency tables: The letters (a, b, c, d) can be helpful for calculations, but can lead to mistakes if not used carefully. UNIVERSITY OF SURREY 14 Risk Reduction Risk reduction refers to the reduction in likelihood of a bad event/outcome occurring due to an intervention or treatment. We can either use an absolute or a relative measure. Absolute risk reduction (ARR) is the absolute difference between the control and experimental group (control event rate minus experimental event rate) ARR = CER − EER Relative risk reduction (RRR) is the proportion by which the treated group improves compared to the control group; it tells you by how much the treatment reduced the risk of a bad outcome relative to the control group CER − EER RRR = CER UNIVERSITY OF SURREY 15 Risk Reduction: Example (Vaccination) From before: CER = 95%, EER = 10% Absolute risk reduction (ARR) CER - EER = 95% - 10% = 85% Interpretation: There is an 85% reduction in the risk of the outcome in the vaccinated group compared with the unvaccinated group. Relative risk reduction (RRR) (CER - EER)/CER = 85%/95% ( = 0.85/0.95) ≈ 89.4% Interpretation: There is an 89.4% reduction in the relative risk of the outcome in the vaccinated group compared with the unvaccinated group. UNIVERSITY OF SURREY 16 Absolute vs. relative risk reduction (ARR vs. RRR) Now suppose we are looking at vaccination against two different diseases, and have calculated the CER and EER for each. % of Animals which develop disease Unvaccinated Vaccinated ARR RRR (i.e. control group) (i.e. treatment group) 0.88 (88%) 0.15 (15%) 0.88 −0.15 Disease 1 0.73 (73%) = 0.83 = 83% (CER) (EER) 0.88 0.088 (8.8%) 0.015 (1.5%) 0.088 −0.015 Disease 2 0.073 (7.3%) = 0.83 = 83% (CER) (EER) 0.088 The two vaccines against different diseases have identical RRR but the ARR benefits have a 10-fold difference. ARR depends on baseline risk (risk of outcome under no treatment conditions i.e. CER), and is a clearer, more practical measure of treatment benefit in many clinical situations RRR removes consideration of baseline risk UNIVERSITY OF SURREY 17 Number needed to treat (NNT) Number needed to treat (NNT) is the number of subjects/animals that need to be treated in order to have one additional successful outcome (equivalently: to prevent one additional bad outcome). NNT can be calculated as the inverse of the absolute risk reduction: 1 write ARR as a decimal NNT = ARR NNT is a useful measure of benefit – it an easily interpretable expression of the effort needed to get a therapeutic result, and is relatively easy to calculate A very small NNT (an NNT that approaches 1) suggests that a favourable outcome occurs in nearly every subject that receives treatment, higher NNTs represent less good treatment The size of the NNT can be weighed up against safety and cost of the treatment UNIVERSITY OF SURREY 18 Number needed to treat (NNT): Example Using previous example: % of Animals which develop disease Unvaccinated Vaccinated ARR RRR NNT Disease 1 88% (CER) 15% (EER) 73% 83% 1/0.73 = 1.37 Disease 2 8.8% (CER) 1.5% (EER) 7.3% 83% 1/0.073 = 13.7 Interpretation: Disease 1: For 1 animal treated with the vaccine, 1 additional animal will benefit Disease 2: For 14 animals treated with the vaccine, 1 additional animal will benefit Note: For NNT, we typically round to the nearest whole number. UNIVERSITY OF SURREY 19 Number needed to treat (NNT) continued A negative NNT implies that the treatment is harmful; we may refer to this as Number needed to harm (NNH), e.g. we could express NNT = -4 as NNH = 4 Number needed to harm (NNH) is the number of subjects/animals that need to be treated in order to have one additional bad outcome A note on good outcomes/events: So far, we have assumed that the event is something bad, e.g. disease. We may want to consider a good event, e.g. recovery from a disease / symptom reduction In this case, we can use: ARR = EER – CER (i.e. switch order of subtraction) and then calculate NNT as normal (see Cyclosporine vs. Prednisolone example in extra material) o Note: May see other ways of handling this, but the most important thing is to keep in mind what is being considered as the “event”, and the direction of interpretation of the measures calculated i.e. do they represent a protective or a harmful effect UNIVERSITY OF SURREY 20 Confidence intervals (CIs) - general We saw in AS1 that a confidence interval is a range of values that indicates the uncertainty around an estimate of a population parameter This parameter could be a population mean or proportion, but confidence intervals can also be calculated for many other estimates, such as a regression parameter, a difference between means, odds ratio, risk ratio, ARR, NNT A confidence interval is typically written as: (lower bound, upper bound) lower upper bound bound The width of a confidence interval indicates the precision of the estimate A wide confidence interval, indicating greater uncertainty (and less precision), is typically due to a small sample size and/or a lot of variability in the data [See extra material for general confidence interval calculation] UNIVERSITY OF SURREY 21 Confidence intervals for ARR/NNT Suppose we have the following ARRs and CIs from two studies investigating the effectiveness of a vaccination against a particular disease: Study 1: ARR = 0.073, 95% CI = (0.060, 0.086) Study 2: ARR = 0.061, 95% CI = (0.020, 0.102) The ARR estimate from the first study is more precise. The 95% CI on an NNT = 1/95% CI on its ARR For example, for study 1 above, we have NNT = 14, 95% CI = (1/0.086, 1/0.06) = (12,17) [See extra material for further example of ARR, NNT and confidence interval calculations] UNIVERSITY OF SURREY 22 Diagnosis LO: Be able to calculate, interpret and explain the terms risk reduction & NNT, absolute risk and relative risk, sensitivity and specificity, predictive values, likelihood ratios, odds, odds ratios, probabilities, pre- and post-test odds LO: Be able to use contingency tables Diagnostic tests Diagnostic tests are tests performed to determine Typical variables: the presence or absence of a specific disease, True status (e.g. disease; condition, or infection in an individual (in this lecture, “positive”/no we will refer to disease from now on) disease; “negative) Diagnostic test validation studies evaluate test's Test result (+ve/-ve) accuracy, comparing its results to a reference gold standard to determine its reliability and clinical utility To estimate statistical measures for diagnostic tests (e.g. sensitivity and specificity), each animal needs to be classified definitively using a “gold-standard” assessment, as well as classified using the diagnostic test being assessed UNIVERSITY OF SURREY 24 Diagnostic test: Example (FeLV) This example will be used to demonstrate the analysis methods seen throughout this section. Disease: FeLV (feline leukemia virus, a common infectious disease in cats) o Categories: viraemic (has virus) vs. non- viraemic (does not have virus) Diagnostic test: IDEXX SNAP test (uses ELISA* technology) o Categories: positive test result vs. negative test result * ELISA = enzyme-linked immunosorbent assay UNIVERSITY OF SURREY 25 Assessing diagnostic tests As for hypothesis tests, there are four possible outcomes: (“Truly” refers to the classification from gold standard (positive = has disease, negative = does not have disease)) False positive: Test positive but True positive: Truly negative (does Test positive and not have disease) Truly positive (has disease) True negative: False negative Test negative and Test negative but Truly negative (does Truly positive (has disease) not have disease) There are a number of methods for assessing the ‘success’ of the test… UNIVERSITY OF SURREY 26 Sensitivity/Specificity and PPV/NPV Note: This section (sensitivity/specificity, PPV/NPV, test thresholds) is a recap of your VMS2008 lecture: Interpretation of diagnostic tests (Week 3) Revisit this lecture for further explanation and examples Key measures for interpreting/assessing the results of a diagnostic test: Sensitivity & specificity status classified by gold standard Sensitivity = The proportion of truly positive cases that are correctly identified as such Specificity = The proportion of truly negative cases that are correctly identified as such the test results are the ‘predicted values’ - want Predictive values to know what proportion of these are correct Positive predictive value (PPV) = Proportion of test positives that are truly positive Negative predictive value (NPV) = Proportion of test negatives that are truly negative UNIVERSITY OF SURREY 27 Test thresholds Many diagnostic tests involve measuring a Simplified illustration: substance on continuous scale, and using a defined cut-off point or “threshold” to decide what results indicate a positive test substance measurement (continuous) Lowering the test threshold increases the sensitivity of the test (and lowers If cut-off is here, then If cut-off is here, then specificity) sensitivity = 100% and sensitivity = 0% Increasing the test threshold increases the specificity = 0% specificity = 100% (all animals test positive) (all animals test negative) specificity of the test (and lowers sensitivity) = animal does not have disease It can be difficult to decide on a cut-off if there is a lot of overlap between test values = animal has disease for animals with and without the disease UNIVERSITY OF SURREY 28 Sensitivity and specificity: Example (FeLV) number of cats that are 𝐭𝐫𝐮𝐥𝐲 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝐀𝐍𝐃 𝐭𝐞𝐬𝐭 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 True positives 𝐒𝐞𝐧𝐬𝐢𝐭𝐢𝐯𝐢𝐭𝐲 = total number of 𝐭𝐫𝐮𝐥𝐲 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 True positives + false negatives a 63 63 = = = ≈ 0.91 = 𝟗𝟏% a+c 63 + 6 69 i.e. 91% of cats with the virus tested positive number of cats that are 𝐭𝐫𝐮𝐥𝐲 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐀𝐍𝐃 𝐭𝐞𝐬𝐭 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐒𝐩𝐞𝐜𝐢𝐟𝐢𝐜𝐢𝐭𝐲 = total number of 𝐭𝐫𝐮𝐥𝐲 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 d 717 717 = = = ≈ 0.98 = 𝟗𝟖% b+d 14 + 717 731 i.e. 98% of cats without the virus tested negative UNIVERSITY OF SURREY 29 PPV and NPV: Example (FeLV) number of cats that are 𝐭𝐞𝐬𝐭 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝐀𝐍𝐃 𝐭𝐫𝐮𝐥𝐲 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 True positives 𝐏𝐏𝐕 = total number of 𝐭𝐞𝐬𝐭 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 True positives + false positives a 63 63 = = = ≈ 0.81 = 𝟖𝟏% a+b 63 + 14 77 i.e. 81% of cats that tested positive had the virus number of cats that are 𝐭𝐞𝐬𝐭 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐀𝐍𝐃 𝐭𝐫𝐮𝐥𝐲 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 𝐍𝐏𝐕 = total number of 𝐭𝐞𝐬𝐭 𝐧𝐞𝐠𝐚𝐭𝐢𝐯𝐞 d 717 717 = = = ≈ 0.99 = 𝟗𝟗% c+d 6 + 717 723 i.e. 99% of cats that tested negative did not have the virus UNIVERSITY OF SURREY 30 Likelihood ratios Likelihood ratios (LRs) are another statistical tool for assessing how good a diagnostic test is, and are derived from sensitivity and specificity The LR is the likelihood that a given test result would be expected in an animal with the disease, compared to the likelihood that that same result would be expected in an animal without the disease A positive likelihood ratio (LR+) measures how much sensitivity more likely a positive test result is in individuals with 𝐋𝐑+ = 1 − specificity a disease compared to those without it A negative likelihood ratio (LR-) measures how much 1 − sensitivity 𝐋𝐑− = more likely a negative test result is in individuals with specificity a disease compared to those without it UNIVERSITY OF SURREY 31 Likelihood ratios: Example (FeLV) Previously we calculated: sensitivity = 0.91 (91%), specificity = 0.98 (98%) sensitivity 0.91304… LR+ = 1 −specificity = 1 − 0.98085… = 47.6783 … ≈ 47.7 Interpretation: A cat with the virus is 47.7 times as likely to test positive than a cat without the virus. 1 −sensitivity 1 −0.91304… LR- = = ≈ 0.09 specificity 0.98085 Interpretation: A cat with the virus is 0.09 times as likely to test negative than a cat without the virus. UNIVERSITY OF SURREY 32 Odds vs. probability Odds and probability are related but distinct concepts in statistics. Probability measures the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain). Odds express the ratio of the probability of success (event occurs) to the probability of failure (event does not occur). For example: probability 0.1 If probability = 0.1 then odds = (= 1:9 = 0.1111) 0.9 1 – probability 2 odds If odds = 2 (i.e. 2:1) then probability = 3 odds + 1 UNIVERSITY OF SURREY 33  Pre- and post-test odds & probabilities We can use the likelihood ratios to calculate the post-test probabilities of having the disease. Pre-test probability is the probability that the animal has the disease, before the test is carried out o Pre-test probability = the prevalence of the disease Pre-test odds is the odds that the animal has the disease, before the test is carried out pre−test probability o Pre-test odds = 1 −pre−test probability Post-test [odds/probability] is the [odds/probability] that the animal has the disease, after the test result has been observed For the post-test probability of the animal having the disease, after positive test result, use LR+. o Post-test odds = pre-test odds x LR post−test odds For the post-test probability of the animal having o Post-test probability = the disease, after a negative test result, use LR-. post−test odds + 1 UNIVERSITY OF SURREY 34 Pre- and post-test odds and probabilities Example (FeLV) Suppose that the population prevalence of FeLV in cats is 10%. We also calculated that LR+ ≈ 47.7. If we look at a cat randomly selected from the population, then: Pre-test probability = prevalence = 10% = 0.1 pre−test probability 0.1 Pre-test odds = = = 0.111111111… ≈ 11% 1 −pre−test probability 0.9 Post-test odds = pre-test odds x LR+ = 0.1111… x 47.67832 = 5.29759… ≈ 5.3 post−test odds 5.29759 Post-test probability = = ≈ 0.84 = 84% post−test odds + 1 5.29759+1 Interpretation: The probability of the cat having FeLV increases from 10% to 84% with a positive test result. (Use LR- to find the probability of having FeLV with a negative test.) Note: A higher prevalence and/or higher likelihood ratio leads to a higher post-test probability. For example, repeating the above with prevalence = 50% given post-test odds = approx. 98% (try it!) UNIVERSITY OF SURREY 35 Aetiology Aetiology = the cause, set of causes, or manner of causation of a disease or condition LO: Be able to calculate, interpret and explain the terms risk reduction & NNT, absolute risk and relative risk, sensitivity and specificity, predictive values, likelihood ratios, odds, odds ratios, probabilities, pre- and post-test odds LO: Be able to use contingency tables Aetiology: Example This example will be used to demonstrate some of the analysis methods seen in this section. Exposure: Exposure might be “risk factors” suspected of causing diseases, or of protecting against it o Categories: yes vs. no Adverse outcome: o Categories: present (yes) vs. absent (no) Note: Aetiological studies can be cohort, cross-sectional or case-control studies. In case-control studies, we refer to those with the outcome of interest as the “cases”, and those without as the “controls”. UNIVERSITY OF SURREY 37 Absolute risk Absolute risk refers to the probability of an outcome occurring in a specific group regardless of any other factors, i.e. the proportion of subjects at risk who have the outcome e.g.: 55 Absolute risk of outcome in all subjects = 100 = 0.55 (55%) 45 [Absolute risk of outcome in exposed group]* = 50 = 0.9 (90%) 10 [Absolute risk of outcome in unexposed group]* = 50 = 0.2 (20%) Relative risk compares the risk of an outcome between different groups. * Analogous to EER/CER, but used in the context of exposure to a risk factor rather than receiving a treatment UNIVERSITY OF SURREY 38 Odds vs. risk Number of subjects/animals with event of interest: 𝑥 Total number of subjects/animals at risk: 𝑛 Risk Odds Based on a proportion Based on a ratio 𝒙 𝒙 Risk = Odds = 𝒏 𝒏−𝒙 Proportion of subjects at risk who Ratio of those with the outcome of interest, have the outcome of interest to those without the outcome of interest To investigate association between risk factor and disease, compare risk/odds of disease between the exposed and unexposed group. Two approaches: relative measures and absolute measures… UNIVERSITY OF SURREY 39 Relative measures : Ratios A relative measure estimates the magnitude of an association between exposure and disease i.e. aetiological strength It indicates how much more likely the exposed group is to develop the disease (or other outcome) than the unexposed group risk in exposed group Like EER/CER in Relative risk (RR) or ‘risk ratio’ = treatment risk in unexposed group context odds in exposed group Odds ratio (OR) = odds in unexposed group UNIVERSITY OF SURREY 40 Relative risk (RR): Example Adverse outcome Risk in exposed group = 45 Present Absent number with outcome in exposed group = number at risk in exposed group 50 45 5 Yes a b Risk in unexposed group = Exposure number with outcome in unexposed group 10 10 40 = No number at risk in unexposed group 50 c d 45 𝑎 risk in exposed group 50 𝑎+𝑏 0.9 Relative Risk = = 10 = 𝑐 = = 4.5 risk in unexposed group 0.2 50 𝑐+𝑑 Interpretation: The risk of the outcome in the exposed group is 4.5 times higher than the risk in the unexposed group. (Outcome is “4.5 times as likely” for exposed group as for unexposed group) UNIVERSITY OF SURREY 41 Odds ratio (OR): Example Adverse outcome Odds in exposed group = 45 Present Absent number with outcome in exposed group = number without outcome exposed group 5 45 5 Yes Odds in unexposed group = a b Exposure number with outcome in unexposed group 10 10 40 = No c d number without outcome in unexposed group 40 45 𝑎 odds in exposed group 5 𝑏 𝑎𝑑 9 Odds ratio = = 10 = 𝑐 = = = 36 odds in unexposed group 𝑏𝑐 0.25 40 𝑑 The odds of the outcome in the exposed group are 36 times the odds in the unexposed group (not “36 times as likely) UNIVERSITY OF SURREY 42  OR and RR: Use and interpretation Odds ratios and risk ratios (for adverse outcomes): A value of 1 indicates there is no difference between exposed and unexposed groups A value > 1 show adverse effect of exposure A value < 1 shows protective effect of exposure o E.g. RR = 1.33 implies the risk in exposed group is 1.33 times higher than risk in unexposed group (33% increase in risk) Agreement between OR an RR will be better for rare outcomes We can calculate confidence intervals (CIs) for risk ratios and odds ratios: A wide CI indicates a low level of precision of the RR/OR estimate, a narrow CI indicates a higher precision of the RR/OR estimate A confidence interval that does not include 1 (the ‘null value’) suggests stronger evidence of an effect For case-control studies: We can calculate odds ratios In most cases, we cannot estimate risk, risk ratio or individual odds (because the prevalence of disease in each group (exposed /unexposed) is usually distorted due to the nature of sampling) UNIVERSITY OF SURREY 43 Absolute risk difference Information on relative measures alone may not be appropriate for studying disease association o E.g. If we are given an increase in risk of 45% (RR = 1.45), we should ask “from what?” Sometimes known as attributable risk, risk difference is an estimate of the amount of risk that is attributable to the exposure of interest Risk difference = risk in exposed group – risk in unexposed group Difference = 0: no difference between groups Difference > 0: risk is greater in exposed group similar to ARR seen earlier Difference < 0: risk is lower in exposed group Attributable (absolute) risk depends on baseline risk UNIVERSITY OF SURREY 44 Prognosis: Survival curves LO: Be able to interpret survival curves Survival data and analysis Prognostic studies in veterinary research may involve the analysis of survival data Survival data (otherwise known as ‘time-to-event’ data) occur where times are recorded from a specific, well defined time origin until the occurrence of some pre-defined event or end-point of interest for a group of subjects (animals) E.g. time from the onset of a disease until death The event of interest is often death, hence the name ‘survival’ data Survival analysis refers to the methods used for analysing survival data UNIVERSITY OF SURREY 46 Features of survival data: Skewness Generally, survival data are positively skewed (usually a lot of subjects experience the event close to the time origin) For example, this histogram summarises the survival time (in weeks) for patients diagnosed with multiple myeloma (time origin = diagnosis). We can see that these data exhibit positive skewness (mean = 24.4 months, median = 15.5 months). https://doi.org/10.4236/ojapps.2020.104010 UNIVERSITY OF SURREY 47 Features of survival data: Censoring The main distinguishing feature of survival data is censoring For each subject in a survival dataset, we have two pieces of information (variables): o Whether the subject experiences the event, or is censored o Survival time, T (time of event of interest or time of censoring) A subject’s recorded survival time is said to be censored if it does not correspond to the time at which the event of interest actually occurs, for example: o The subject drops out of the study before experiencing the event o The event of interest does not occur during the observation period / follow-up e.g. subject is still alive at the end of follow-up period We need to model the data in a way that takes account of censoring UNIVERSITY OF SURREY 48 Survivor function and the Kaplan-Meier estimate When summarising survival data, we are usually very interested in the survivor function, S(t), which measures the probability that an individual will experience the event of interest at or beyond some time t (equivalently, that they will survive until time t) If there are no censored subjects in a dataset, we can use a simple, empirical estimate of the survivor function: number of subjects with survival times ≥ t Ŝ 𝑡 = total number of subjects in the dataset In the far more common scenario of censored data, we need an alternative approach which takes the censoring into account: the Kaplan-Meier estimate is the most common choice (The Kaplan-Meier estimate is based on events that happen within each of a series of time intervals constructed using the event times) UNIVERSITY OF SURREY 49 Interpreting survival curves Kaplan-Meier estimate of the survivor function The plot of the Kaplan-Meier estimate of the survivor for time until death function is called a Kaplan-Meier survival curve These two Y-axis: Probability that a subject survives beyond subjects had just over 50 time t (X-axis: time t) weeks of follow-up The tick marks represent censored times Remembering that survival data are usually positively skewed, we are often interested in the median when estimating average survival time The median survival time is the smallest time for which Ŝ 𝑡 ≤ 0.5 We can read the median survival time from the Time until death (t weeks) plot UNIVERSITY OF SURREY 50 Comparing survival curves Often, we want to compare survivor Survival rate of domesticated dogs on function estimates for different groups of different diets (fabricated example) subjects/animals, to see if there is a difference in survival between the groups As a visual comparison, we can plot separate survival curves for each group In this example, the estimated survivor function decreases more rapidly over time for the Diet 2 group than for the Diet 1 group The median survival time is lower for the Diet 2 group than for the Diet 1 group https://www.graphpad.com/guides/survival-analysis (approx. 3 year vs. approx. 7 years) UNIVERSITY OF SURREY 51 Summary and Resources Summary LOs Analyses covered, in the context of types of studies they are usually seen in: Be able to calculate, interpret Therapy and explain the terms risk CER and EER reduction & NNT, absolute risk Risk reduction (absolute and relative) and relative risk, sensitivity and NNT (and NNH) specificity, predictive values, Diagnosis likelihood ratios, odds, odds Sensitivity and specificity ratios, probabilities, pre- and Predictive values (NPV & PPV) post-test odds Likelihood ratios Pre- and post-test odds, and pre- and post-test probabilities Be able to use contingency tables Aetiology Odds and odds ratios Be able to interpret survival Risk, absolute risk and relative risk curves Prognosis studies Survival curves UNIVERSITY OF SURREY 53 Resources - Applied Statistics 1 & 2 Holmes, M. and Cockcroft, P. (2008) Handbook of Veterinary Clinical Research, Blackwell Publishing MASA statistical test selector and practice scenarios for choosing a test: https://surreylearn.surrey.ac.uk/d2l/le/lessons/215610/folders/1862937 MASA statistics glossary: https://surreylearn.surrey.ac.uk/d2l/le/lessons/215610/folders/2725689 scribbr.co.uk - concise, user-friendly summaries of statistical topics (e.g. https://www.scribbr.co.uk/stats/descriptive-statistics-explained/) LAERD Statistics: clear and concise web pages on a wide range of statistical concepts and tests - search for ‘LAERD’ followed by the concept/test you are looking for in your search engine e.g. “LAERD descriptive statistics” / “LAERD paired t-test” (pages include SPSS guides where relevant e.g. for statistical tests) MASA is available for drop-ins and appointments, including advice on any more complex statistical methods you may come across in your literature review https://surreylearn.surrey.ac.uk/d2l/le/lessons/215610/lessons/1856863 MASA open-to-all workshop: Choosing a statistical test, Wednesday 27th November, 2-2.50pm UNIVERSITY OF SURREY 54 Extra material Number needed to treat (NNT) Example: Cyclosporine vs Prednisolone “In a paper by Olivry et al. (2002), the use of oral cyclosporine Treatment was compared to the use of prednisolone for the treatment of canine atopic dermatitis in a randomised control trial.”* Cyclosporine Prednisolone Results of the trial: Experimental Control; usual treatment treatment Prednisolone: Reduced pruritis in 10/14 dogs; CER = 71.42857143% ≈ 71.4 % Outcome: Yes 10 10 Reduction Cyclosporine – Reduced pruritis in 10/13 in pruritis dogs; EER = 76.92307692% ≈ 76.9 % (good No 3 4 Absolute Risk Reduction (ARR) = EER – outcome) CER (since event is good) ≈ 5.5% EER = 10/13 CER = 10/14 NNT = 1/0.055 = 18.2 dogs ≈ 76.9%* ≈ 71.4%* Interpretation: for 18 dogs treated with cyclosporine * Example quoted from: Holmes, M. and Cockcroft, P. (2008) Hand- rather than prednisolone, 1 additional dog will benefit. book of Veterinary Clinical Research, Blackwell Publishing, pp. 110 56 Confidence intervals General formula for constructing a 95% confidence interval: 95% confidence interval = point estimate ± (1.96 x standard error of point estimate) We use a sample This is the critical value Standard error: statistic calculated and depends on the dependent on from our sample confidence level that we variability within to estimate the set. the sample, and population sample size For a normal parameter we are distribution, the critical interested in value for a 95% confidence interval is approximately 1.96. UNIVERSITY OF SURREY 57 Confidence interval for a single proportion Then the 95% CI for the proportion is (using rounded value for simplicity): 𝑃 ∗ (1 −𝑃) CI = P ± ( 1.96 x ) 𝑛 standard error (SE) of proportion 102 Absolute risk = ≈ 0.016 (1.6%) 6255 0.016 ∗ (0.984) = 0.016 ± ( 1.96 x ) 6255 n = number of subjects P = sample proportion – here this is absolute risk = 0.016 ± 0.0031 UNIVERSITY OF SURREY 58  Confidence interval for ARR How do we calculate a confidence interval for absolute risk reduction (ARR)? General 95% confidence interval = point estimate ± (1.96 x standard error of point estimate) 𝐶𝐸𝑅 (1−𝐶𝐸𝑅) 𝐸𝐸𝑅 (1−𝐸𝐸𝑅) 95% confidence interval for the ARR = ARR ± ( 1.96 x + ) 𝑁𝑜. 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑎𝑛𝑖𝑚𝑎𝑙𝑠 𝑁𝑜. 𝑜𝑓 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑎𝑛𝑖𝑚𝑎𝑙𝑠 standard error (SE) of ARR UNIVERSITY OF SURREY 59 Confidence intervals for ARR/NNT Example: Cyclosporine vs Prednisolone The 95% CI on an NNT = 1/95% CI on its ARR Example: Cyclosporine vs Prednisolone ARR = 0.055 (5.5%) 95% CI on ARR = - 0.274 to 0.384 NNT = 1/ARR = 18.2 ≈ 18 dogs So 95% CI on NNT = 1/-0.274 to 1/0.384 ≈ -3.65 to 2.60 Why are the confidence intervals for the ARR and NNT are so wide? NNH 4 dogs NNT 3 dogs Small number of dogs in the trial (27 dogs) UNIVERSITY OF SURREY 60 Confidence intervals for ARR/NNT Example: Cyclosporine vs Prednisolone - calculation 𝐶𝐸𝑅 (1−𝐶𝐸𝑅) 𝐸𝐸𝑅 (1−𝐸𝐸𝑅) 95% confidence interval for the ARR = ARR ± ( 1.96 x + 𝑁𝑜. 𝑜𝑓 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑎𝑛𝑖𝑚𝑎𝑙𝑠 ) 𝑁𝑜. 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑎𝑛𝑖𝑚𝑎𝑙𝑠 The ARR we calculated 0.714 (1−0.714) 0.769 (1−0.769) 95% confidence interval for the ARR = 5.5% ± ( 1.96 x + ) 14 13 0.204024 0.177639 = 5.5% ± ( 1.96 x + ) 14 13 = 5.5% ± ( 1.96 x 0.02825053846 ) ≈ 0.329 = 5.5% ± 32.9% = ( - 27.4% , 38.4%) or ( - 0.274, 0.384) UNIVERSITY OF SURREY 61 Nomogram for calculating post-test probabilities Switching between pre- and post-test probabilities can be done easily using a nomogram. E.g. Using FeLV example with prevalence = 50%: Pre-test probability = 50% (0.5) Positive likelihood ratio = 47.7 Then from nomogram: Post-test probability = 98% (as before) And we could work back the other way! UNIVERSITY OF SURREY 62