basic statistics

Basic Statistics

Tim Horn AIDSmeds.com/ATAC

[email protected]

Overview

Statistics definedMean, Median & ModeRangesStandard DeviationStandard ErrorConfidence IntervalsHypotheses and P-valuesRisk Ratios

Statistics Defined

The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from samples Population: a complete set of people to be studied.

e.g. All people living with HIV Sample: a smaller part of that population

Using research involving a sample, statistics can draw conclusions or make predictions about what may happen in the larger population

Mean, Median & Mode

Mean, Median & Mode

Three kinds of “averages” in statistics: Mean Median Mode

All numbers in a set of data fall within a range

Mean Example

J Acquir Immune Defic Syndr. 2010 Apr 1;53(4):456-63.Long-term efficacy and safety of the HIV integrase inhibitor raltegravir in patients with limited

treatment options in a Phase II study.Gatell JM, Katlama C, Grinsztejn B, Eron JJ, Lazzarin A, Vittecoq D, Gonzalez CJ, Danovich RM, Wan H,

Zhao J, Meibohm AR, Strohmaier KM, Harvey CM, Isaacs RD, Nguyen BY; Protocol 005 Team.AbstractBACKGROUND: Raltegravir in combination therapy has demonstrated potent suppression of HIV-1 with a

favorable safety profile. This report provides 96-week efficacy and safety data from Protocol 005, a Phase II study. METHODS: HIV-infected patients with very limited treatment options and failing antiretroviral therapy were randomized to raltegravir 200, 400, or 600 mg or placebo b.i.d., plus optimized background therapy for >or=24 weeks; all patients were then offered open-label raltegravir 400 mg b.i.d. Efficacy measurements included changes in viral load and CD4 count from baseline and percent of patients with HIV-1 RNA <400 and <50 copies/mL. RESULTS: One hundred and thirty-three patients received raltegravir and 45 received placebo. No dose-dependent differentiation in the safety or antiviral activity of raltegravir was observed during the double-blind phase. For the combined raltegravir groups, mean change in viral load from baseline was -1.60 log10 copies/mL at week 48 and -1.38 log10 copies/mL at week 96 (observed failure approach). At week 48, HIV-1 RNA levels were <400 copies/mL in 68% of raltegravir recipients and <50 copies/mL in 55%; these levels were maintained in 55% and 48% of raltegravir recipients, respectively, at week 96 (noncompleter = failure). There were few discontinuations of raltegravir (4%) due to adverse events. CONCLUSIONS: In patients with limited treatment options, raltegravir with OBT had a potent and sustained antiretroviral effect and was generally well tolerated through 96 weeks.

Mean

The mean is the average you're used to, where you add up all the numbers and then divide by the number of numbers

11 injection drug users (IDUs) who tested positive for HIV at the following ages: 19, 24, 32, 32, 27, 29, 21, 32, 36, 39, 31

(19, 24, 32, 32, 27, 29, 21, 32, 36, 39, 31) ÷ 11 = 29

Median Example

AIDS Patient Care STDS. 2010 Apr 8. [Epub ahead of print]

Hepatitis B Virus Drug Resistance in HIV-1-Infected Patients Taking Lamivudine-Containing Antiretroviral Therapy.

Wongprasit P, Manosuthi W, Kiertiburanakul S, Sungkanuparph S.

1 Department of Medicine, Faculty of Medicine, Ramathibodi Hospital, Mahidol University , Bangkok, Thailand .

A cross-sectional study was conducted in HIV-1-infected patients receiving lamivudine-containing antiretroviral therapy (ART) to determine the prevalence and risk factors of hepatitis B virus drug resistance (HBV-DR). HBV DNA and HBV genotypic resistance test were performed. Patients were categorized into two groups: with and without HBV-DR. There were 84 patients with a mean age (standard deviation [SD]) of 42.2 (10.2) years and 77% were males. Median (range) duration of ART and lamivudine use was 46 (3-177) and 40 (3-140) months, respectively. Median (range) CD4 cell count was 352 (49-790) cells/mm(3). Of all, 19 (23%) had HBV-DR with a median (range) HBV DNA of 2.56 x 10(7) (2.54 x 10(3)-11 x 10(7)) IU/mL. In univariate analysis, there were no differences in age, gender, ART regimen, liver function test, anti-HBc antibody, anti-HCV antibody between the two groups. Patients with HBV-DR had a higher proportion of positive HBeAg (68.4% versus 3.8%, p < 0.001). In multivariate analysis, positive HBeAg (odds ratio [OR) 16.64; 95% confidence interval [CI], 3.31-83.60] and duration of lamivudine use [per 6-month increment, OR 1.24; 95% CI, 1.06-1.36] were significant risk factors for HBV-DR. All 19 patients with HBV-DR had lamivudine resistance with the mutations as follows: M204V/I (95%), L180M/A181T (95%), L80V/I (47%), V173L (32%), and N236T (21%). Of these, 95%, 84%, 84%, and 0% of patients had HBV-DR to telbivudine, entecavir, adefovir, and tenofovir, respectively. HBV-DR is common in HBV/HIV-1 coinfected patients receiving lamivudine-containing ART without tenofovir. Positive HBeAg and longer duration of lamivudine use are risk factors for HBV-DR. In addition to lamivudine resistance, cross-resistance to other anti-HBV drugs is also frequently observed.

Median

The median is the middle value (age), so they need to be listed in chronological order 19, 21, 24, 27, 29, 31, 32, 32, 32, 36, 39

There are 11 numbers in the list, so the middle one will be the 5th number: 19, 21, 24, 27, 29, 31, 32, 32, 32, 36, 39 The median is 31

Mode

The number (age) that occurs most often in the group 19, 21, 24, 27, 29, 31, 32, 32, 32, 36, 39 32 is the mode

Rarely used in HIV epidemiology or treatment research

Ranges and Centiles

Ranges

Median provides no information about range of values or how values are grouped around the median The range

Difference between highest and lowest number The Interquartile range

Data divided into quarters: • First quartile (25th centile)• Second quartile (50th centile) • Third quartile (75th centile)

Ranges

19, 21, 24, 27, 29, 31, 32, 32, 32, 36, 39 The largest value in the list is 39, and the

smallest is 19, so the range is 39 – 19 = 20

Interquartile Range

19, 21, 24, 27, 29, 31, 32, 32, 32, 36, 39 First quartile is 24 Second quartile is 31 (median) Third Quartile is 32

IQR: 32-24=8

IQR Example

Infection. 2010 Mar 29. [Epub ahead of print]

Safety and Efficacy of a Saquinavir-Containing Antiretroviral Regimen in Previously ART-Naïve or Pretreated but Protease Inhibitor-Naïve HIV-Positive Patients.

Knechten H, Stephan C, Mosthaf FA, Jaeger H, Lutz T, Cargnico A, Stoehr A, Koeppe S, Mayr C, Schewe K, Wolf E, Wellmann E, Tappe A.

Practice Center Blondelstrasse (PZB), Blondelstr. 9, 52062, Aachen, Germany, [email protected].

Abstract

BACKGROUND: The RAINBOW survey is a multinational observational study assessing the tolerability and efficacy of ritonavir-boosted saquinavir (SQV/r), using the 500-mg film-coated SQV formulation, in routine clinical practice. This analysis presents data from the German subgroup of antiretroviral therapy (ART)-naïve and pretreated but protease inhibitor (PI)-naïve patients. METHODS: This was a multicenter, prospective, open-label, 48-week observational cohort study. Tolerability assessments included changes in liver enzymes and lipid levels from baseline to week 48. Efficacy assessments included changes in the proportion of patients with HIV-1 RNA <50 and <400 copies/ml, and changes in CD4 cell count from baseline to week 48. RESULTS: The analysis included 275 ART-naïve and 179 pretreated but PI-naïve patients. The proportion of ART-naïve patients achieving <50 copies/ml by 48 weeks was 53.1% by intent-to-treat (ITT) analysis and 67.3% using last observation carried forward (LOCF) analysis. In pretreated but PI-naïve patients, the proportions achieving <50 copies/ml by 48 weeks were 53.1% (ITT) and 70.4% (LOCF). The median increase in CD4 count at week 48 was +174 cells/mm(3) (interquartile range [IQR] 86, 265) in the ART-naïve group and +100 cells/mm(3) (IQR 0, 209) in the pretreated but PI-naïve group (p < 0.01 for both; LOCF). Drug-related adverse events were reported in 7.6% of ART-naïve and 2.8% of pretreated but PI-naïve patients. Treatment with SQV/r was stopped in 21.5% of ART-naïve and 17.9% of pretreated but PI-naïve patients (due to side effects in 3.3% and 2.8%, respectively). There were no clinically relevant changes in liver enzyme levels. Overall, the total cholesterol, triglyceride, low-density lipoprotein, and high-density lipoprotein levels increased to week 48, although the levels remained within normal ranges in the majority of patients. CONCLUSIONS: The results of this observational cohort study of treatment with the 500-mg tablet formulation of SQV are consistent with high efficacy and tolerability results seen in controlled studies of SQV/r. This analysis confirms that SQV/r is effective and well tolerated in ART-naïve and pretreated but PI-naïve patients in 'real-world' clinical settings.

Standard Deviation

Standard Deviation (SD)

IQR indicates the variation of data where the median average is used

Standard deviation (SD) is used when the mean average is used Indicates the difference between a group of

values and their mean The larger the SD, the more spread out the

values


Represented as number of SDs away from mean

Approximately 68% of the scores will fall between one SD above the mean and one SD below the mean; 95% of all scores will fall between 2 SDs above and below the mean; 99.7% of scores will fall between 3 SDs above and below the mean


Example: The average height for adult men in the U.S. is about 70 inches, with a SD of around 3 in.

This means that most men (68%) have a height within 3 in of the mean – one standard deviation, whereas almost all men (about 95%) have a height within 6 in of the mean (64–76 in) – 2 standard deviations.

If the standard deviation were zero, then all men would be exactly 70 in (178 cm) high.

If the standard deviation were 20 in, then men would have much more variable heights, with a typical range of about 50 to 90 in.

Three standard deviations account for 99.7% of the sample population being studied.

Standard Deviation (SD) Example

AIDS Patient Care STDS. 2010 Apr 8. [Epub ahead of print]

Hepatitis B Virus Drug Resistance in HIV-1-Infected Patients Taking Lamivudine-Containing Antiretroviral Therapy.

Wongprasit P, Manosuthi W, Kiertiburanakul S, Sungkanuparph S.

1 Department of Medicine, Faculty of Medicine, Ramathibodi Hospital, Mahidol University , Bangkok, Thailand .

Abstract

A cross-sectional study was conducted in HIV-1-infected patients receiving lamivudine-containing antiretroviral therapy (ART) to determine the prevalence and risk factors of hepatitis B virus drug resistance (HBV-DR). HBV DNA and HBV genotypic resistance test were performed. Patients were categorized into two groups: with and without HBV-DR. There were 84 patients with a mean age (standard deviation [SD]) of 42.2 (10.2) years and 77% were males. Median (range) duration of ART and lamivudine use was 46 (3-177) and 40 (3-140) months, respectively. Median (range) CD4 cell count was 352 (49-790) cells/mm(3). Of all, 19 (23%) had HBV-DR with a median (range) HBV DNA of 2.56 x 10(7) (2.54 x 10(3)-11 x 10(7)) IU/mL. In univariate analysis, there were no differences in age, gender, ART regimen, liver function test, anti-HBc antibody, anti-HCV antibody between the two groups. Patients with HBV-DR had a higher proportion of positive HBeAg (68.4% versus 3.8%, p < 0.001). In multivariate analysis, positive HBeAg (odds ratio [OR) 16.64; 95% confidence interval [CI], 3.31-83.60] and duration of lamivudine use [per 6-month increment, OR 1.24; 95% CI, 1.06-1.36] were significant risk factors for HBV-DR. All 19 patients with HBV-DR had lamivudine resistance with the mutations as follows: M204V/I (95%), L180M/A181T (95%), L80V/I (47%), V173L (32%), and N236T (21%). Of these, 95%, 84%, 84%, and 0% of patients had HBV-DR to telbivudine, entecavir, adefovir, and tenofovir, respectively. HBV-DR is common in HBV/HIV-1 coinfected patients receiving lamivudine-containing ART without tenofovir. Positive HBeAg and longer duration of lamivudine use are risk factors for HBV-DR. In addition to lamivudine resistance, cross-resistance to other anti-HBV drugs is also frequently observed.

Standard Error

Standard Error (SE)

The standard error (SE) is important in describing how well the sample mean represents the true population mean Every random sample will give a slightly different

estimation of the whole population SE gives you a measure of how precise your sample

mean is compared to the true population mean Calculated by the standard deviation divided by the

square root of the mean Depends on the sample size. As the sample size gets

larger, then variability gets smaller, yielding more precise measurement of the truth

Standard Error ExampleCROI Paper 972

Life Expectancy of Persons at the Time of Initiating cART in High-income CountriesRobert Hogg and Antiretroviral Cohort CollaborationBC Ctr for Excellence in HIV/AIDS and Simon Fraser Univ, Vancouver, Canada

Background: To characterize changes in mortality and life expectancy among HIV+ persons initiating combination ART (cART).

Methods: The Antiretroviral Cohort Collaboration (ART-CC) is a multinational cohort study of ART-naive patients initiating cART in Europe and North America. Patients were included in this analysis if they were on cART for at least a year. The primary endpoint was all-cause mortality. Abridged life tables were constructed to estimate life expectancies among ART-naive persons at the time of initiating cART in 3 periods 1996-1999, 2000-2002, and 2003-2005. These tables were stratified by gender, baseline CD4 cell count, and history of injection drug use. For this exercise, the expectation of life at age 20 years was reported and refers to the average number of years remaining to be lived by those initiating cART at that age. Potential years of life lost from 20 to 65 years and crude death rates were also calculated for this exercise.

Results: A total of 14,993, 9895, and 3614 patients initiated and were on cART for at least 1 year in 1996-1999, 2000-2002, and 2003-2005, respectively. A total of 1531 (5.4%) deaths were observed in this population during the study period with crude death rates decreasing from 45.1 deaths per 1000 person years in 1996-1999 to 27.8 deaths per 1000 person-years in 2003-2005. Potential years of life lost per 1000 person-years also decreased over the same time interval from 1796.7 to 1296.2. Life expectancy at exact age 20 years increased from 24.3 years (standard error, SE, 0.8) to 33.2 (SE 0.8) during this time span. Life expectancy levels were comparable for men and women at 33.5 years (SE 1.2) and 33.0 years (SE 1.3), respectively in 2003-2005. Patients with a history of injection drug use had significantly lower life expectancies than those from other transmission groups (28.2 years, SE 1.0, vs 34.7 years, SE 0.9, in 2003-2005). During 2003-2005, life expectancy decreased at lower baseline CD4 counts, ranging from 38.3 years (SE 0.8) for those with baseline CD4 counts of ≥350 cells/mm3 to 30.9 (SE 1.3) for those with baseline CD4 counts of <200 cells/mm3.

Conclusions: The average number of years remaining to be lived by those initiating cART at age 20 years were approximately half those observed among the general population in these countries. In the United States, life expectancy at age 20 years was 58.3 years in 2003.

Confidence Intervals

Confidence Intervals (CIs)

CIs used to estimate how far away the population mean is likely to be from the sample mean, with a degree of certainty

A CI is an estimate of the spread between the lowest likely result (lower confidence limit) and the highest likely result (upper confidence limit) of a study The true result of the study probably lies somewhere

within this CI The smaller the spread of the confidence interval, the

more precise the result is likely to be


Most research studies display results using 95% CI i.e., there is a 95% chance that the true study

result lies between these two confidence limits

Uses standard error The mean plus or minus two times the standard

error


In a hypothetical example, it may be reported that "The estimated number of people living with HIV among a group of IDUs was 18.6% with a 95% CI of 12.9-24.0.” This means that the study investigators are

95% sure that the true prevalence lies somewhere between the two confidence limits of 12.9% and 24.0%

Confidence Intervals (CIs) Example

N Engl J Med. 2009 Dec 3;361(23):2209-20. Epub 2009 Oct 20.

Vaccination with ALVAC and AIDSVAX to prevent HIV-1 infection in Thailand.

Rerks-Ngarm S, Pitisuttithum P, Nitayaphan S, Kaewkungwal J, Chiu J, Paris R, Premsri N, Namwat C, de Souza M, Adams E, Benenson M, Gurunathan S, Tartaglia J, McNeil JG, Francis DP, Stablein D, Birx DL, Chunsuttiwat S, Khamboonruang C, Thongcharoen P, Robb ML, Michael NL, Kunasol P, Kim JH; MOPH-TAVEG Investigators.

Abstract

BACKGROUND: The development of a safe and effective vaccine against the human immunodeficiency virus type 1 (HIV-1) is critical to pandemic control. METHODS: In a community-based, randomized, multicenter, double-blind, placebo-controlled efficacy trial, we evaluated four priming injections of a recombinant canarypox vector vaccine (ALVAC-HIV [vCP1521]) plus two booster injections of a recombinant glycoprotein 120 subunit vaccine (AIDSVAX B/E). The vaccine and placebo injections were administered to 16,402 healthy men and women between the ages of 18 and 30 years in Rayong and Chon Buri provinces in Thailand. The volunteers, primarily at heterosexual risk for HIV infection, were monitored for the coprimary end points: HIV-1 infection and early HIV-1 viremia, at the end of the 6-month vaccination series and every 6 months thereafter for 3 years. RESULTS: In the intention-to-treat analysis involving 16,402 subjects, there was a trend toward the prevention of HIV-1 infection among the vaccine recipients, with a vaccine efficacy of 26.4% (95% confidence interval [CI], -4.0 to 47.9; P=0.08). In the per-protocol analysis involving 12,542 subjects, the vaccine efficacy was 26.2% (95% CI, -13.3 to 51.9; P=0.16). In the modified intention-to-treat analysis involving 16,395 subjects (with the exclusion of 7 subjects who were found to have had HIV-1 infection at baseline), the vaccine efficacy was 31.2% (95% CI, 1.1 to 52.1; P=0.04). Vaccination did not affect the degree of viremia or the CD4+ T-cell count in subjects in whom HIV-1 infection was subsequently diagnosed. CONCLUSIONS: This ALVAC-HIV and AIDSVAX B/E vaccine regimen may reduce the risk of HIV infection in a community-based population with largely heterosexual risk. Vaccination did not affect the viral load or CD4+ count in subjects with HIV infection. Although the results show only a modest benefit, they offer insight for future research. (ClinicalTrials.gov number, NCT00223080.) 2009 Massachusetts Medical Society

Putting it All Together

CD4 Cell Counts in Random Sample of 50 PLWHIV

Hypotheses and P-Values

Hypotheses and P-values

A study begins with a null hypothesis and an alternative hypothesis Null: Isentress plus OBT is no more effective than

OBT alone in keeping VL <50 copies for 48 wks Alternative Isentress plus OBT more effective

than OBT alone in keeping VL <50 copies for 48 wks

Researchers always hope to reject the null hypothesis and prove the alternative hypothesis

Hypotheses and P-values

After data are collected, statistician calculates p-value Determines whether the study supports the null or

alternative hypotheses P-value is the probability that these results would

occur if there was truly no difference between the groups -- that is, how likely the results would have been observed purely by chance

P-values are between 0 and 1; the closer to zero the less likely the null hypothesis is true

Closely tied to the confidence interval

Hypothesis and P-values

65% of pts. receiving Isentress/OBT have VLs <50 after 48 weeks, vs. 35% receiving OBT w/out Isentress

P-value is 0.01 (P=0.01) Represents 1 in 100 probability that the null hypothesis is

true – a very small, highly significant (statistical) difference. A statistical significance level below 5% is

considered to be “good enough.” However, if the findings are likely to have very important consequences for medical interventions or public policy, a lower p-value is demanded (e.g., p < 0.01).

Hypothesis and P-values

P-values and CIs often reported together The p-value is a single number that guides

whether or not to reject the null hypothesis The 95% CI provides a range of plausible

values for describing the underlying population If CIs do not overlap, differences are statistically

significant If CIs do overlap, differences may not be statistically

significant

Risk Ratios

Relative Risk (RR)

Risk of an event (or of developing a disease) relative to exposure. Relative risk is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group

Generally used in randomized controlled trials and cohort studies

Not the same as absolute risk!

Relative Risk (RR)

A relative risk of 1 indicates a lack of difference between the two groups in terms of risk (e.g., RR=1.0)

A relative risk less than 1 indicates the trait has a lesser likelihood of being expressed in the experimental group than in the control group (e.g., RR=0.7)

A relative risk greater than 1 indicates the trait has a greater likelihood of being expressed in the experimental group than in the control group (e.g., RR=1.16)

Relative Risk – D:A:D Example

The RR of chronic kidney disease (CKD) increases by 16% per year among those taking regimens containing tenofovir (e.g., Atripla) Compared to the same person’s underlying risk

of CKD (and never exposed to tenofovir)

Relative Risk – D:A:D Example

If a person has an underlying absolute risk of 0.50 percent for developing CKD within the next 12 months one year of tenofovir exposure would lead to a 16% relative increase in the absolute risk of CKD – i.e. an absolute risk of 0.58 percent (0.5x1.16)

If a person has an underlying absolute risk of 20% for developing CKD in the next 12 months, one year of tenofovir exposure would lead to an increase in this absolute of CKD to 23.2%

Hazard Ratio

Broadly equivalent to relative risk (RR); useful when the risk is not constant with respect to time. It uses information collected at different times.

The term is typically used in the context of survival over time.

If the HR is 0.5 then the relative risk of dying in one group is half the risk of dying in the other group

Odds Ratio (OR)

Like RR and HR, the OR compares the likelihood of an event or treatment effect between two groups

Often used in case-controlled and retrospective studies

When events are rare the OR is analagous to the relative risk (RR), but as event rates increase the OR and RR diverge

Risk Ratios: Putting it All Together

In a randomized trial of 100 patients given a study drug to prevent PCP (50 given placebo and 50 given study drug)

PCP No PCP Total

Placebo 10 40 50

Study drug

5 45 50


Absolute Risk of PCP if given placebo: 10/50 = 0.2 or 20%

Absolute Risk of PCP if given study drug: 5/50 = 0.1 or 10%

Relative risk of PCP if given study drug: 0.1 / 0.2 = 0.5 or 50%

Difference in absolute risk of PCP if given study drug: 0.1 - 0.2 = -0.1 or 10% reduction

PCP No PCP Total

Placebo 10 40 50

Study drug 5 45 50


Odds of PCP if given placebo: 10:40 = 1:4 or 0.25

Odds of PCP if given study drug: 5:45 = 1:9 or 0.11

Odds ratio of PCP if given study drug: 0.11 / 0.25 = 0.44

PCP No PCP Total

Placebo 10 40 50

Study drug 5 45 50

Other Terms

Incidence: The number of new cases of a particular disease or condition over a defined period of time (e.g., new HIV infections occurring in 2009).

Prevalence: The overall number of cases of a particular disease over a defined period of time (e.g., total number of people living with HIV in 2009).

Presenting Data

A variety of graph styles can be used to present data

Used to convey the importance of data quickly Graphs should be simple and easy

Presenting Data: Graphs

Pie Chart

Compare part of a whole at a given point in time.

New HIV/AIDS cases in 2005, by race

Line Chart

To find and compare trends (changes over time)

Estimated number of deaths among adults with AIDS, 1985–1999, United States

Bar Graph

Women Aged 18 and Older who Have Ever Been Tested for HIV, by Race/Ethnicity, 2007

Used to compare frequencies (percentage of women testing for HIV) in different categories (race/ethnicity)

Bar GraphAdults Aged 18 and Older who Have Ever Been Tested for HIV, by Age and Sex, 2007

Used to compare two frequencies (percentage of men and women testing for HIV) in different categories (age)

Histogram

AIDS cases, by age and sex, reported 1981–2000, United States

A bar chart representing a frequency distribution; heights of the bars represent observed frequencies

Box-and-Whiskers Plot

Total abdominal fat among patients on ARV treatment with lipodystrophy, on ARV treatment w/out lipodystrophy and HIV-negative controls

A convenient way of graphically depicting five sets of data: the smallest observation (sample minimum), first quartile, second quartile (median), third quartile, and largest observation (sample maximum).

basic statistics

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