Introduction to Statistics

Dr Nikola Grubor

2024-10-13

What is statistics for?

• Applied scientific method
• Discovery of treatment effect (drug, intervention)
• Predicting risk (e.g., thrombosis)
• Medical decision making (diagnostic test accuracy)

Hypertension treatment effect

What is the recommendation based on? Does it make sense?

Among patients at high risk for cardiovascular events but without diabetes, targeting a systolic blood pressure of less than 120 mm Hg, as compared with less than 140 mm Hg, resulted in lower rates of fatal and nonfatal major cardiovascular events and death from any cause, although significantly higher rates of some adverse events were observed in the intensive-treatment group.

Risk prediction

Is pulse pressure or systolic pressure responsible for most of the risk of hemorrhagic stroke?

How much better is ultrasound at detecting gallstones than CT?

Laboratory values

Does this patient with this lab value have a disease risk that requires further investigation?

Course

• Lesson (\(\gt\) 50% \(\rightarrow\) opens quiz)
• \(\text{Points} = \text{lesson} + \text{quiz (time limited)} + \text{colloquium} + \text{exam}\)
• Exam (practical and theoretical part; computer-based assessment in the classroom)

Course structure

Nedelja	Tema
1	Introduction to Medical Statistics
2	Data Sorting
3	Statistical description of data
4	Probability and probability distributions
5	Population and Sample
6	Testing Hypotheses About Population Means and Proportions
7	Frequency Hypothesis Testing
8	Rank Hypothesis Testing
9	Correlation
10	Regression / Colloquium
11	Data, information and knowledge
12	Health information system
13	Diagnostic accuracy measures
14	Medical decision-making
15	Bioinformatics

Critical state of science in medicine

Richard Horton, M.D., Lancet Editor-in-Chief

“The case against science is straightforward: much of the scientific literature, perhaps half, may simply be untrue. Afflicted by studies with small sample sizes, tiny effects, invalid exploratory analyses, and flagrant conflicts of interest, together with an obsession for pursuing fashionable trends of dubious importance, science has taken a turn towards darkness.”

1,693,059 publications on PubMed in 2023!

How often do results replicate?

After 2000 the FDA requires preregistration

Definition

Statistics is the science of decision making

in conditions of uncertainty

Basic terms

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flowchart LR
  A[Base Set] --> B[Sample]
  B --> C[Unit of observation]
  C --> D[Variables]
  D --> E[Measurement]
  E <--> TT["Accuracy\nPrecision\nReliability\nValidity"]
  TT <--> F
  E <--> TP["Data Type"]
  TP <--> F
  E --> F[Data]
  F --> G[Analysis]
  G --> H[Conclusion]
  H -->|generalization|A

The research question

Defining the research question (estimand) is something that is mostly forgotten when designing a study.

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flowchart TD
  I[Reseach question] -.-> A
  A[Basic set] --> B[Sample]
  B --> C[Unit of observation]
  C --> D[Variable]

Estimand je tačna mera koju želimo na naučimo (rizik, učestalnost, prosek, prosečni kauzalni efekat terapije). Pretstavlja jos precizniju definiciju istraživačkog pitanja.

Example: research question

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flowchart TD
  I[Reseach question] -.-> A
  A[Basic set] --> B[Sample]
  B --> C[Unit of observation]
  C --> D[Variable]

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flowchart TD
  I["Is the degree of degeneration\nof the lumbosacrum on X-ray\nassociated with pain?"] -.-> A
  A[Patients with lumboischialgia] --> B[n patients]
  B --> C[Patient]
  C --> D["Degree of degeneration, pain"]

The gingival swelling and dusky color just above two of the teeth indicate hemorrhage into the gums of this patient with poor dentition. The gingival abnormalities of scurvy occur only in the presence of teeth, which presumably provide portals of entry for microbes into the gums. One hypothesis suggests that vitamin C deficiency impairs neutrophil-mediated killing of bacteria, leading to chronic gingivitis, which is then complicated by bleeding from the fragile vessels characteristic of scurvy.

In this example, the perifollicular hyperkeratotic papules are quite prominent, with surrounding hemorrhage. These lesions have been misinterpreted as “palpable purpura,” leading to the mistaken clinical diagnosis of vasculitis.

James Lind’s research

James Lind (1716-1794)

HMS Salisbury (1747)

The first medical database

study_id	treatment	gum_rot_d6	skin_sores_d6	fit_for_duty_d6
Notes: Abridged database of original data collected by James Lind; d6, on day 6.
001	cider	2_moderate	2_moderate	0_no
002	cider	2_moderate	1_mild	0_no
003	dilute_sulfuric_acid	1_mild	3_severe	0_no
004	dilute_sulfuric_acid	2_moderate	3_severe	0_no
005	vinegar	3_severe	3_severe	0_no
006	vinegar	3_severe	3_severe	0_no
007	sea_water	3_severe	3_severe	0_no
008	sea_water	3_severe	3_severe	0_no
009	citrus	1_mild	1_mild	0_no
010	citrus	0_none	0_none	1_yes
011	purgative_mixture	3_severe	3_severe	0_no
012	purgative_mixture	3_severe	3_severe	0_no

Measurement

• The method of measurement is very important
• Accuracy, reliability, validity, missing data

Picasso’s drawing of a dog.

Nama je ocigledan pas, ali kako izmeriti koliko je ovo pas? Treba nam neka skala. Primeri: koliko je teska neka bolest (DM, HTA, stadijum Ca), znanje (ocigledno losa mera), itd. Merenja mogu da budu i opasna: nekad smo merili lobanje (frenologija).

Patient responses to blood pressure measurement

SPRINT trial ima ozbilje kritike.

In clinical practice, BP is often measured with little attention to quality control and may overestimate average SBP by as much as 10 to 15 mm Hg compared with values obtained using guideline recommended methods that have been employed in most landmark antihypertensive drug treatment trials.

Data types

Data consists of variables. Variables represent a measure of some phenomenon.

• Qualitative (describe the phenomenon)
• Quantitative (assign a numerical value)

Important

“What is your favorite food?”, to which the answers are: grapes, chocolate, carbonara, pizza and mango. This data is not numerical; but we often assign it numbers (1 = grapes, 2 = chocolate, etc.), however in this case the numbers serve as labels, not real values.

Mera nije samo rec, ver pojam koji se odnosi na proces dodavanja vrednosti nekim opservacijama. Starost merimo vremenom, tezinu kilogramima, bol proizvoljnom skalom 0-10, antitela koncentracijom, rizik verovatnoćom, itd.

Qualitative data

What is your favorite subject?	Number of students
Pathology	105
Statistics	20
Pathophysiology	4

Note

The students’ answers are qualitative, but they can be turned quantitative by summarizing.

Number types lead to data types

• Binary numbers (Yes/No; 1/0; TRUE/FALSE)
• Integers (\(\mathbb Z\)) (“Agreement on a scale of 1 to 7?”)
• Reals (\(\mathbb R\)) (most common, \(3.12...\), \(9.81...\))

Discontinuous (discrete) and continuous measurements

Discontinuous (discrete) measurements are qualitative or can be expressed with integers (whole numbers)

Example

Number of births: 3, no sense in talking about 3.5 births!

Continuous measurements are expressed by real numbers

Example

When we measure weight in \(kg\) we usually round the numbers so they look whole, but it sill makes sense to talk about more precise measurements (\(70 \; \text{kg}\), \(70.3467 \; \text{kg}\)).

The most important takeaway from this is that some kinds of statistics don’t make sense on some kinds of data. Zip code basically serves as a label for a different region. For this reason, it wouldn’t make sense to talk about the average zip code, for example.

Measurement scales (1)

Variables must have at least two values (otherwise they would be constants).

Variable values have different relationships:

• Identity: Each value is unique.
• Size (magnitude): Some values are larger than others.
• Equal intervals: The difference between 1 and 2 is the same size as between 19 and 20.
• Absolute zero: The scale has a zero that is meaningful.

Zasto su bitne? Govore nam koje simplifikacije imaju smisla.

Measurement scales (2)

• Nominal scale (identity; 1 - Diabetes, 2 - Hypertension)
• Ordinal scale (identity and size; 1 - Primary education, 2 - Secondary school)
• Interval scale (identity, magnitude, and equal intervals; Celsius, Fahrenheit)
• Ratio scale (all + absolute zero; Kelvin, height, weight)

Allowed operations based on data scale

Scale	Equal/Not Equal	>/<	+/-	Multiplication/Division
Nominal	OK
Ordinal	OK	OK
Interval	OK	OK	OK
Ratio	OK	OK	OK	OK

These constraints imply certain kinds of statistics. Statistics that simply count values (mode), can be calculated on any of the variable types. Statistics based on ordering or ranking of values (median), require that the value at least be on an ordinal scale. Finally, statistics that involve adding up values (such as the average, or mean), require that the variables be at least on an interval scale.

Nominal data

• Unique values
• They have no minimum or maximum
• No natural ordering

Ordinal data

• Discrete (integers)
• Defined minimum and maximum
• Defined order
• The “distance” between the categories is unknown

“Ponekad” je veće od “Nikad”, ali koliko veće je ponekad od nikad? Koliko je 2 na skali bola veće od 1? Odgovor nije 1, nego neki subjektivna mera. Koliko neko voli ovaj čas, za 1-2 nije teško, ali 4-5 je teško.

Numerical data

• Whole numbers are discontinuous
• Continuous data can be arbitrarily precise

Warning

Ordinal and discrete numerical data can look similar. The difference between them is that the numbers in ordinal data are only labels, not real values. Continuous and discontinuous data may look similar if rounded (50kg, 60kg, 75kg), but we always have the option to display them more precisely.

Trombociti su prebrojivi, ali nekad se mere kontinuiranim merama kao npr. spektrometrijom. Vezbe iz biohemiije (kolorimetrija).

Guess the data type?

• ABO
• Pain (NRS 0-10 scale)
• M/F
• Stage (I-IV)
• TA (mmHg)
• Heart rate (bpm)
• Age

• Nominal
• Ordinal
• Nominal (binary)
• Ordinal
• Numerical (cont.)
• Numerical (discrete)*
• Numerical (cont.)

What about time?

Uvek je bolje tretirati vreme kao kontinuirano (mada zavisi od pitanja, individualni efekti vs. trend).

Excel for data entry

Tidy data principles (tidydata)

ugh…

Importing a database into the EZR

• Download and import satisfaction with health care.xlsx into EZR
• Load KVS.xlsx without overwriting the previous database

Important

Select Sheet1 (this is where the data is usually located) when importing the database. Click View to check that the correct database is loaded!

What is a code book and what is it for?

Istraživačka baza podataka praćena je šifarnikom koji sadrži informacije o nazivu i redosledu varijabli, opisu varijabli, mernim jedinicama, i skali merenja (nominalni, ordinalni, numerički). Osim toga u šifarniku su, za kategorijalne varijable, date informacije o broju kategorija i kodovima tih kategorija.

Common mistakes in data collection

Granular data (disaggregation)

Granular Data (merging)

Column repetition (unrecognized variables)

Repeated measurements (easier typing)

Exercise: different bases

Design a database for:

• Comparing the values of systolic pressure between diabetics and healthy people?
• Comparing glucose values between the same patients before and after therapy?
• Recording CRP measured serially at the following times: 1h, 12h, 24h?

Hard (optional)

• Follow-up patients to find out who had a relapse after surgery?
• Measure time to thrombosis between patients treated aggressively and conservatively.
• Measure time to thrombosis while monitoring d-Dimer values serially for 3 days in a row.

Solution: diabetics and healthy people

Gluc	Group
7.2	DM
5.1	Healthy
5.9	Healthy
4.7	Healthy
11.1	DM

Solution: before and after

gluc_pre	gluc_post
7.2	6.1
5.1	5.1
5.9	5.5
4.7	5.9
11.1	7.1

Solution: measurement series

Variant 1:

ID	CRP	Time
1	0.62	1h
1	0.51	12h
1	0.56	24h
2	0.43	1h
2	0.42	12h

Variant 2:

ID	CRP_1h	CRP_12h	CRP_24h
1	0.70	0.67	0.54
2	0.66	0.41	0.59
3	0.76	0.49	0.53

Missing data

patient	intervention	outcome
Doe Jones	1	7
NA	2	10
NA	3	NA
Mary Smith	1	4

• Leave blank (possible errors)
• Enter NA