Download presentation

Presentation is loading. Please wait.

Published byRosalyn Wheeler Modified about 1 year ago

1
Descriptive Epidemiology Dr. KANUPRIYA CHATURVEDI

2
How we view the world….. Pessimist: The glass is half empty. Pessimist: The glass is half empty. Optimist: The glass is half full. Optimist: The glass is half full. Epidemiologist: As compared to what? Epidemiologist: As compared to what?

3
Epidemiology is...

4
"The worst taught course in Medical school." Medical Student

5
Epidemiology is... "The science of making the obvious obscure." Clinical Professor

6
Epidemiology is... "The science of long division.... I'=[(480)(log2)(10E6)]/[(9.1)(0.955po)+0.45 n]" Statistician

7
Definition of Epidemiology* "The STUDY of the DISTRIBUTION and DETERMINANTS of HEALTH- RELATED STATES in specified POPULATIONS, and the application of this study to CONTROL of health problems." *Last, J.M A Dictionary of Epidemiology, 2nd ed.

8
Epidemiology: Definition Dynamic study of the Determinants Occurrence Distribution Control Pattern Of health and disease in a population

9
Epidemiology Epidemiology EPI DEMO LOGOS Upon,on,befall People,population,man the Study of The study of anything that happens to people “That which befalls man”

10
Definition of Epidemiology A quantitative basic science, built on a working knowledge of probability, statistics and sound research methods. A method of causal reasoning, based on developing and testing biologically plausible hypothesis pertaining to occurrence and prevention of morbidity and mortality. A tool for public health action to promote and protect the public's health based on science, causal reasoning, and a dose of practical common sense.

11
Epidemiology is a Quantitative Discipline Measures of frequency Measures of frequency Counts and rates Counts and rates Measures of association Measures of association Relative risk Relative risk Odds ratio Odds ratio Statistical inference Statistical inference P-value P-value Confidence limits Confidence limits

12
Clinician Epidemiologist Patient’s diagnostician Investigations Diagnosis Therapy Cure Community’s diagnostician Investigations Predict trend Control Prevention

13
Epidemiology Describes Describes health events cause and risk factors of disease clinical pattern of disease Identify syndromes Identify control and/or preventive measures Identify control and/or preventive measures

14
So, Epidemiology Is the basic science of public health Is the basic science of public health Provides insight regarding the nature, causes, and extent of health and disease Provides insight regarding the nature, causes, and extent of health and disease Provides information needed to plan and target resources appropriately Provides information needed to plan and target resources appropriately

15
Kinds of Epidemiology Descriptive Descriptive Analytic Analytic Experimental Experimental Further studies to determine the validity of a hypothesis concerning the occurrence of disease. Deliberate manipulation of the cause is predictably followed by an alteration in the effect not due to chance Study of the occurrence and distribution of disease

16
Overview of epidemiologic design strategies Descriptive Descriptive Populations{Correlational studies} Individual Case report Case series Cross sectional studies Analytic studies Analytic studies Observational Case control Cohort Retrospective Prospective Interventional/Experimental Randomized controlled trial Field trial Clinical trial

17
Descriptive vs. Analytic Epidemiology Descriptive Descriptive Used when little is known about the disease Used when little is known about the disease Rely on preexisting data Rely on preexisting data Who, where, when Who, where, when Illustrates potential associations Illustrates potential associationsAnalytic Used when insight about various aspects of disease is available Used when insight about various aspects of disease is available Rely on development of new data Rely on development of new data Why Why Evaluates the causality of associations Evaluates the causality of associations Both are important!

18
Descriptive Studies Relatively inexpensive and less time-consuming than analytic studies, they describe, Relatively inexpensive and less time-consuming than analytic studies, they describe, Patterns of disease occurrence, in terms of, Patterns of disease occurrence, in terms of, Who gets sick and/or who does not Who gets sick and/or who does not Where rates are highest and lowest Where rates are highest and lowest Temporal patterns of disease Temporal patterns of disease Data provided are useful for, Data provided are useful for, Public health administrators (for allocation of resources) Public health administrators (for allocation of resources) Epidemiologists (first step in risk factor determination) Epidemiologists (first step in risk factor determination)

19
Descriptive Epidemiology Correlational studies Correlational studies Case reports Case reports Case series Case series Cross sectional studies Cross sectional studies

20
Correlational Studies (Ecological Studies) Uses measures that represent characteristics of entire populations Uses measures that represent characteristics of entire populations It describes outcomes in relation to age, time, utilization of services, or exposures It describes outcomes in relation to age, time, utilization of services, or exposures ADVANTAGES ADVANTAGES We can generate hypotheses for case-control studies and environmental studies We can generate hypotheses for case-control studies and environmental studies We can target high-risk populations, time-periods, or geographic regions for future studies We can target high-risk populations, time-periods, or geographic regions for future studies

21
Correlational Studies LIMITATIONS LIMITATIONS Because data are for groups, we cannot link disease and exposure in individual Because data are for groups, we cannot link disease and exposure in individual We cannot control for potential confounders We cannot control for potential confounders Data represent average exposures rather than individual exposures, so we cannot determine a dose-response relationship Data represent average exposures rather than individual exposures, so we cannot determine a dose-response relationship Caution must be taken to avoid drawing inappropriate conclusions, or ecological fallacy Caution must be taken to avoid drawing inappropriate conclusions, or ecological fallacy

22
Patterns of disease Occurrence : Correlation of Population statistics Ecologic ( correlation ) studies Ecologic ( correlation ) studies Used as first step in determining association plot : disease (population) burden [ Y axis ] vs. prevalence of “risk factor” [ X axis ] e.g. smoking vs. lung cancer -- correlation coefficient : r ; + 1 to correlation coefficient : r ; + 1 to -1 Quantifies linear relationship between exposure & disease Quantifies linear relationship between exposure & disease

23
Case Reports (case series) Report of a single individual or a group of individuals with the same diagnosis Report of a single individual or a group of individuals with the same diagnosis Advantages Advantages We can aggregate cases from disparate sources to generate hypotheses and describe new syndromes Example: hepatitis, AIDS Example: hepatitis, AIDS Limitations Limitations We cannot test for statistical association because there is no relevant comparison group Based on individual exposure {may simply be coincidental}

24
Case report/Case series(contd.) Important interface between clinical medicine & epidemiology Important interface between clinical medicine & epidemiology Most common type of studies published in medical journals{1/3 rd of all} Most common type of studies published in medical journals{1/3 rd of all} e.g. Frisbee finger, break dancing neck e.g. Frisbee finger, break dancing neck AIDS ~ b/w oct1980-may81, 5 cases of P.carinii pneumonia were diagnosed among previously healthy young homosexual males in L.A. AIDS ~ b/w oct1980-may81, 5 cases of P.carinii pneumonia were diagnosed among previously healthy young homosexual males in L.A.

25
Cross-Sectional Studies (prevalence studies) Measures disease and exposure simultaneously in a well-defined population Measures disease and exposure simultaneously in a well-defined population Advantages Advantages They cut across the general population, not simply those seeking medical care They cut across the general population, not simply those seeking medical care Good for identifying prevalence of common outcomes, such as arthritis, blood pressure or allergies Good for identifying prevalence of common outcomes, such as arthritis, blood pressure or allergies Limitations Limitations Cannot determine whether exposure preceded disease Cannot determine whether exposure preceded disease It considers prevalent rather than incident cases, results will be influenced by survival factors It considers prevalent rather than incident cases, results will be influenced by survival factors Remember: P = I x D Remember: P = I x D

26
Cross-Sectional Studies Can be used as a type of analytic study for testing hypothesis, when; Current values of exposure variables are unalterable over time Represents value present at initiation of disease E.g. eye colour or blood group If risk factor is subject to alterations by disease, only hypothesis formulation can be done

27
The epidemiologic approach: Steps to public health action MEASURES Counts Times Rates Risks/Odds Prevalence METHODS Design Conduct Analysis Interpretation ALTERNATIVE EXPLANATIONS Chance Bias Confounding INFERENCES Epidemiologic Causal ACTION Behavioural Clinical Community Environmental DESCRIPTIVE What (case definition) Who (person) Where (place) When (time) How many (measures) ANALYTIC Why (Causes) How (Causes)

28
Key questions Why now? Why now? Why here? Why here? Why in this group? Why in this group?

29
Descriptive Epidemiology Study of the occurrence and distribution of disease Terms: Time Place Person

30
What are the three categories of descriptive epidemiologic clues? □ Person: Who is getting sick? □ Person: Who is getting sick? □ Place: Where is the sickness occurring? □ Place: Where is the sickness occurring? □ Time: When is the sickness occurring? □ Time: When is the sickness occurring? PPT = person, place, time PPT = person, place, time

31
Time Secular Periodic Seasonal Epidemic

32
Secular Trend The long-time trend of disease occurrence

33
Tetanus – by year, USA, During 2000, a total of 35 cases of tetanus were reported. The percentage of cases among persons aged years Has increased in the last decade. Note: A tetanus vaccine was first available in 1933.

34
Possible Reasons for Changes in Trends Artifactual Artifactual Errors in numerator due to Errors in numerator due to Changes in the recognition of disease Changes in the recognition of disease Changes in the rules and procedures for classification of causes of death Changes in the rules and procedures for classification of causes of death Changes in the classification code of causes of death Changes in the classification code of causes of death Changes in accuracy of reporting age at death Changes in accuracy of reporting age at death Errors in the denominator due to error in the enumeration of the population Errors in the denominator due to error in the enumeration of the population

35
Possible Reasons for Changes in Trends (cont.) Real Real Changes in age distribution of the population Changes in age distribution of the population Changes in survivorship Changes in survivorship Changes in incidence of disease resulting from Changes in incidence of disease resulting from Genetic factors Genetic factors Environmental factors Environmental factors

36
Other phrases Cyclic trends ~ recurrent alterations in occurrence, interval or frequency of disease Secular cyclicity Levels of immunizations Build up of susceptibles e.g. Hep A-7 yr cycle,Measles-2yr cycle Short term cyclicity Chickenpox,salmonella(yearly basis)

37
Periodic Trend Temporal interruption of the general trend of secular variation

38
Whooping Cough - Four-monthly admissions,

39
Seasonal A cyclic variation in disease frequency by time of year & season. A cyclic variation in disease frequency by time of year & season. Seasonal fluctuations in, Environmental factors Occupational activities Recreational activities

40
Seasonal Trend Pneumonia-Influenza Deaths – By year,

41
Epidemic An increase in incidence above the expected in a defined geographic area within a defined time period

42
Endemic, Epidemic and Pandemic Endemic - The habitual presence (or usual occurrence) of a disease within a given geographic area Epidemic - The occurrence of an infectious disease clearly in excess of normal expectancy, and generated from a common or propagated source Pandemic - A worldwide epidemic affecting an exceptionally high proportion of the global population Number of Cases of Disease Time

43
Time clustering Time Place Cluster/disease cluster A group of cases occur close together & have a well aligned distribution pattern {in terms of time and place} A group of cases occur close together & have a well aligned distribution pattern {in terms of time and place} Cluster analysis-used for rare or special disease events. Cluster analysis-used for rare or special disease events.

44
Time/Place clustering analysis using the Poisson model {Poisson spatial/nearest neighbor distribution} Poisson probability distribution is an inferential statistics probability measure. Poisson probability distribution is an inferential statistics probability measure. Describes objects/events as they are distributed geographically. Describes objects/events as they are distributed geographically. Geographical area divided into a series of equal square areas. Geographical area divided into a series of equal square areas. Randomization i.e. each case has equal probability of falling into each square. Randomization i.e. each case has equal probability of falling into each square. If clustering occurs, probability of cause-effect relationship goes up & vice versa. If clustering occurs, probability of cause-effect relationship goes up & vice versa.

45
Place Diagnosis is Made Contact occurred between agent and host Source became infected Geographic AreaExample Action Level Home – Patient ill Restaurant – Food Eaten Farm – Eggs Infected Investigation Control Prevention

46

47
Person AgeHobbies SexPets OccupationTravel Immunization statusPersonal Habits Underlying disease Stress MedicationFamily unit Nutritional statusSchool Socioeconomic factorsGenetics CrowdingReligion

48
Descriptive epidemiology : Patterns of Disease Occurrence distribution of disease in populations numerator ( “event” count ) / denominator ( group “at risk” ) distribution of disease in populations numerator ( “event” count ) / denominator ( group “at risk” ) by “person” : age, race / ethnicity, gender, occupation, education, marital status, genetic marker, sexual preference by “person” : age, race / ethnicity, gender, occupation, education, marital status, genetic marker, sexual preference by “place” : residence (urban vs. rural), worksite, social event by “place” : residence (urban vs. rural), worksite, social event by “time” : week, month, year ; sporadic, seasonal, trends --- incubation period ; latency by “time” : week, month, year ; sporadic, seasonal, trends --- incubation period ; latency

49
Sources of information Census data Census data Vital statistical records Vital statistical records Employment health examinations Employment health examinations Clinical records from hospitals Clinical records from hospitals National figures on food consumption, medications, health events etc National figures on food consumption, medications, health events etc

50
Epidemiologic ( scientific ) Approach 1. Identify a PROBLEM : clinical suspicion ; case series ; review of medical literature 1. Identify a PROBLEM : clinical suspicion ; case series ; review of medical literature 2. Formulate a HYPOTHESIS ( asking the right question ) ; good hypotheses are: Specific, Measurable, and Plausible 2. Formulate a HYPOTHESIS ( asking the right question ) ; good hypotheses are: Specific, Measurable, and Plausible 3. TEST that HYPOTHESIS ( assumptions vs. type of data ) 3. TEST that HYPOTHESIS ( assumptions vs. type of data ) 4. always Question the VALIDITY of the result(s) : Chance ; Bias ; and Causality 4. always Question the VALIDITY of the result(s) : Chance ; Bias ; and Causality

51
Epidemiologic Study: threats to Validity Epidemiologic Study: threats to Validity Chance : role of random error in outcome measure(s) ( p - value ; power of the study and the confidence interval ) --- largely determined by sample size Bias : role of systematic error in outcome measure(s) Selection bias - subjects not representative Selection bias - subjects not representative Information bias - error(s) in subject data / classification Information bias - error(s) in subject data / classification Confounding - 3rd variable (causal) assoc. w/ both X and Y Confounding - 3rd variable (causal) assoc. w/ both X and Y

52
What is a hypothesis? An educated guess An educated guess an unproven idea an unproven idea based on observation or reasoning, that can be proven or disproven through investigation. based on observation or reasoning, that can be proven or disproven through investigation.

53
What goes into a hypothesis? Characteristics of the disease Characteristics of the disease The illness The illness Established modes of transmission Established modes of transmission Distribution Distribution In time In time By place By place By person By person

54
Hypothesis formulation 4 methods {derived from 5 canons of inductive reasoning by John Stuart Mill} 4 methods {derived from 5 canons of inductive reasoning by John Stuart Mill} Method of difference Method of agreement Method of concomitant variation Method of analogy

55
Measures Morbidity: Refers to the presence of disease in a population Morbidity: Refers to the presence of disease in a population Mortality: Refers to the occurrence of death in a population Mortality: Refers to the occurrence of death in a population

56
Methods for Measuring How do we determine disease frequency for a population? Rate = Frequency of defined events in specified population for given time period Rate = Frequency of defined events in specified population for given time period Rates allow comparisons between two or more populations of different sizes or of a population over time Rates allow comparisons between two or more populations of different sizes or of a population over time

57
Compute Disease Rate Number of persons at risk = 5,595,211 Number of persons with disease = 17,382 Rate = 17,382 persons with heart disease 5,595,211 persons = heart disease / resident / year

58
Rates Rates are usually expressed as integers and decimals for populations at risk during specified periods to make comparisons easier heart disease / resident / year x 100,000 = heart disease / 100,000 residents / year

59
Prevalence vs. Incidence Prevalence is the number of existing cases of disease in the population during a defined period. Prevalence is the number of existing cases of disease in the population during a defined period. Incidence is the number of new cases of disease that develop in the population during a defined period. Incidence is the number of new cases of disease that develop in the population during a defined period.

60
Incidence Incidence rate is a measure of the probability of the event among persons at risk. Incidence rate is a measure of the probability of the event among persons at risk.

61
Incidence Rates Population denominator: Population denominator: IR = # new cases during time period X K specified population at risk

62
Example (Incidence Rate) During a six-month time period, a total of 53 nosocomial infections were recorded by an infection control nurse at a community hospital. During this time, there were 832 patients with a total of 1,290 patient days. What is the rate of nosocomial infections per 100 patient days?

63
Mortality Rates A special type of incidence rate A special type of incidence rate Number of deaths occurring in a specified population in a given time period Number of deaths occurring in a specified population in a given time period

64
Use of Mortality rates Mortality rates are used to estimate disease frequency when… Mortality rates are used to estimate disease frequency when… incidence data are not available, case-fatality rates are high, goal is to reduce mortality among screened or targeted populations

65
Mortality Rates: Examples Crude mortality: death rate in an entire population Crude mortality: death rate in an entire population Rates can also be calculated for sub-groups within the population Rates can also be calculated for sub-groups within the population Cause-specific mortality: rate at which deaths occur for a specific cause Cause-specific mortality: rate at which deaths occur for a specific cause

66
Mortality Rates: Examples Case-fatality: Rate at which deaths occur from a disease among those with the disease Case-fatality: Rate at which deaths occur from a disease among those with the disease Maternal mortality: Ratio of death from childbearing for a given time period per number of live births during same time period Maternal mortality: Ratio of death from childbearing for a given time period per number of live births during same time period

67
Mortality Rates: Examples Infant mortality: Rate of death for children less than 1 year per number of live births Infant mortality: Rate of death for children less than 1 year per number of live births Neonatal mortality: Rate of death for children less than 28 days of age per number of live births Neonatal mortality: Rate of death for children less than 28 days of age per number of live births

68
Prevalence Prevalence: Existing cases in a specified population during a specified time period (both new and ongoing cases) Prevalence: Existing cases in a specified population during a specified time period (both new and ongoing cases) Prevalence is a measure of burden of disease or health problem in a population Prevalence is a measure of burden of disease or health problem in a population

69
Prevalence Prevalence: The number of existing cases in the population during a given time period. PR =# existing cases during time period population at same point in time Prevalence rates are often expressed as a percentage.

70
Factors Influencing Prevalence Increased by: Longer duration of the disease Prolongation of life of patients without cure Increase in new cases (increase in incidence) In-migration of cases Out-migration of healthy people In-migration of susceptible people Improved diagnostic facilities (better reporting) Decreased by: Shorter duration of disease High case-fatality rate from disease Decrease in new cases (decrease in incidence) In-migration of healthy people Out-migration of cases Improved cure rate of cases

71
Basic Measures of Association Relative risk& odds ratio Relative risk& odds ratio We often need to know the relationship between an outcome and certain factors (e.g., age, sex, race, smoking status, etc.) We often need to know the relationship between an outcome and certain factors (e.g., age, sex, race, smoking status, etc.) Used to guide planning and intervention strategies Used to guide planning and intervention strategies

72
2 x 2 contingency table for Calculation of Measures of Association Outcome ExposurePresentAbsentTOTAL Presentaba+b Absentcdc+d TOTALa+cb+da+b+c+d Note: “Exposure” is a broad term that represents any factor that may be related to an outcome.

73
Relative Risk Ratio of the incidence rates between two groups Ratio of the incidence rates between two groups Can only be calculated from prospective studies (cohort studies) Can only be calculated from prospective studies (cohort studies) Interpretation Interpretation RR > 1: Increased risk of outcome among “exposed” group RR > 1: Increased risk of outcome among “exposed” group RR < 1: Decreased risk, or protective effects, among “exposed” group RR < 1: Decreased risk, or protective effects, among “exposed” group RR = 1: No association between exposure and outcome RR = 1: No association between exposure and outcome

74
Calculation of Relative Risk incidence rate among exposed incidence rate among exposed RR = incidence rate among non-exposed incidence rate among non-exposed

75
Calculation of Relative Risk Outcome Outcome ExposurePresentAbsentTOTAL Presentaba+b Absentcdc+d TOTALa+cb+da+b+c+d Relative Risk =

76
Relative Risk Case Study Birth Weight Smoking status <2500 g >2500 g TOTAL Smoker Non-smoker TOTAL Smoking and low birth weight

77
Answers to Relative Risk Case Study 1. Incidence of LBW among smokers 1. Incidence of LBW among smokers 2. Incidence of LBW among non-smokers 2. Incidence of LBW among non-smokers 3. Relative risk for having a LBW baby among smokers versus non-smokers 3. Relative risk for having a LBW baby among smokers versus non-smokers

78
Understanding Probability and Odds Probability: Chance or risk of an event occurring (a proportion) Probability: Chance or risk of an event occurring (a proportion) Probability= no. of times an event occurs Probability= no. of times an event occurs no. of times an event can occur no. of times an event can occur Odds: ratio of the probability of an event occurring to the probability of an event not occurring Odds: ratio of the probability of an event occurring to the probability of an event not occurring Odds = P/(1-P) Odds = P/(1-P)

79
Calculation of Odds Ratio Outcome Outcome ExposurePresentAbsentTOTAL Presentaba+b Absentcdc+d TOTALa+cb+da+b+c+d Odds Ratio =

80
Odds Ratio The odds ratio (OR) is a ratio of two odds. The odds ratio (OR) is a ratio of two odds. The OR can be calculated for all three study designs The OR can be calculated for all three study designs Cross-sectional Case-control Cohort.

81
Various approaches to Odds ratio Cross product/odds ratio Cross product/odds ratio 2 x 2 contingency table (ad/bc) Prevalence odds ratio Prevalence odds ratio cross sectional studies Exposure odds ratio( odds of exposure in diseased vs. nondiseased) Exposure odds ratio( odds of exposure in diseased vs. nondiseased) In rare cases or exotic diseases Disease odds/Rate odds ratio(o dds of getting a disease if exposed or unexposed) Disease odds/Rate odds ratio(o dds of getting a disease if exposed or unexposed) Cohort & cross sectional Risk odds ratio Risk odds ratio Cross sectional,cohort & case control

82
Odds Ratio For cohort & cross sectional studies: OR is a ratio of the odds of the outcome in exposed persons to the odds of the outcome in non- exposed persons. For cohort & cross sectional studies: OR is a ratio of the odds of the outcome in exposed persons to the odds of the outcome in non- exposed persons. For case-control studies: OR is a ratio of the odds of exposure in cases to the odds of exposure in controls. For case-control studies: OR is a ratio of the odds of exposure in cases to the odds of exposure in controls. Provides an estimate of the relative risk when the outcome is rare Provides an estimate of the relative risk when the outcome is rare

83
Interpretation of Odds Ratio OR > 1: Increased odds of exposure among those with outcome OR > 1: Increased odds of exposure among those with outcome OR < 1: Decreased odds, or protective effects, among those with outcome OR < 1: Decreased odds, or protective effects, among those with outcome OR = 1: No association between exposure and outcome OR = 1: No association between exposure and outcome

84
Keeping the Terms Straight “Risk ratio” = “relative risk” “Risk ratio” = “relative risk” “Relative odds” = “odds ratio” “Relative odds” = “odds ratio” Remember – the key is recognizing the terms “risk” and “odds” Remember – the key is recognizing the terms “risk” and “odds”

85
Appropriateness of Measures Remember that the relative risk can only be calculated in prospective studies Remember that the relative risk can only be calculated in prospective studies Odds ratio can be calculated for any design Odds ratio can be calculated for any design Cohort / prospective Cohort / prospective Case-control Case-control Cross-sectional Cross-sectional

86
Inference The relative risk and odds ratio provide the magnitude of difference between some factor and an outcome The relative risk and odds ratio provide the magnitude of difference between some factor and an outcome How do we know if the magnitude is statistically significant? How do we know if the magnitude is statistically significant?

87
Confidence Intervals A confidence interval is a range of values that is likely (e.g., 95%) to contain the true value in the underlying population A confidence interval is a range of values that is likely (e.g., 95%) to contain the true value in the underlying population

88
The 10 Steps of Outbreak Investigation Prepare for field work Prepare for field work Establish the existence of an outbreak Establish the existence of an outbreak Verify the diagnosis Verify the diagnosis Define & identify cases Define & identify cases Perform descriptive epidemiology Perform descriptive epidemiology Develop hypotheses Develop hypotheses Perform analytic epidemiology Perform analytic epidemiology Refine hypotheses & conduct additional studies Refine hypotheses & conduct additional studies Implement control & prevention measures Implement control & prevention measures Communicate findings Communicate findings

89
Objectives of Descriptive Epidemiology To evaluate trends in health and disease and allow comparisons among countries and subgroups within countries To evaluate trends in health and disease and allow comparisons among countries and subgroups within countries To provide a basis for planning, provision and evaluation of services To provide a basis for planning, provision and evaluation of services To identify problems to be studied by analytic methods and to test hypotheses related to those problems To identify problems to be studied by analytic methods and to test hypotheses related to those problems

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google