Understanding Latent Class Analysis in Research

Latent Class Analysis (LCA) is a person-centered approach that categorizes individuals based on underlying differences. This method links observed behaviors to categorical variations, providing insights into groupings within data sets.

• Latent Class Analysis
• Research Methodology
• Statistical Modeling
• Data Analysis

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1. Introduction to Latent Class Analysis Part II Dr Oliver Perra o.perra@qub.ac.uk

2. Summary: Latent Class Analysis (LCA) Person centred-approach A mixture of individuals assigned to different categories (classes) Measurement model: Observed behaviours are causally related to underlying categorical differences

3. Latent Class Analysis (LCA): Formal definitions A A = Low Mood a can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). B B = Lack of Pleasure b can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). C C = Sleep Problems c can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). D D = Fatigue; d can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). x can be: 1 or 2. X X = Latent class

4. Latent Class Analysis (LCA): Formal definitions A A = Low Mood acan be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). B B = Lack of Pleasure b can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). C C = Sleep Problems c can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). D D = Fatigue; d can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). x can be 1 or 2. X X = Latent class ??????=?? ??|? ??|? ??|???|?

5. Latent Class Analysis (LCA): Formal definitions A A = Low Mood acan be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). B B = Lack of Pleasure b can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). C C = Sleep Problems c can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). D D = Fatigue; d can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). x can be 1 or 2. X X = Latent class ??????=?? ??|? ??|? ??|???|? ??= 1.0 ??|?= ??|?= ??|?= ??|?= 1.0

6. Latent Class Analysis (LCA): Formal definitions A A = Low Mood acan be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). B B = Lack of Pleasure b can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). C C = Sleep Problems c can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). D D = Fatigue; d can be: 1 (Never) ; 2 (Sometimes); 3 (Most of the time). x can be 1 or 2. X X = Latent class ??????=?? ??|? ??|? ??|???|? ??=3; ?=3; ?=3; ?=3; ?=1=? ?=1 ??=3|?=1 ??=3|?=1 ??=3|?=1??=3|?=1

7. Example of Data ID Low Mood Lack Pleasure Sleep Problems Fatigue 101 1 1 2 1 102 3 3 2 3 103 1 2 1 1 104 2 1 3 2

8. Latent Class Analysis (LCA) Parameters A A = Low Mood acan be: B B = Lack of Pleasure b can be: 1 to 3. C C = Sleep Problems c can be: 1 to 3. D D = Fatigue d can be: 1 to 3. X X = Latent class x can be: 1 or 2. 1 (Never) ; 2 (Sometimes); 3 (Most of the time). Class membership probability ??=1 and??=2 with constraint: ??=1 +??=2 = 1.0 ??????=?? ??|? ??|? ??|???|? ??= ??|?= ??|?= ??|?= ??|?= 1.0

9. Example of Data & Output ID Low Mood Lack Pleasure Sleep Problems Fatigue Class membership probability ??=1 and ??=2 ??= 1.0 ??=1 +??=2 = 1.0 101 1 1 2 1 102 3 3 2 3 103 1 2 1 1 104 2 1 3 2 Latent class 0.1254 C1 or else: ??=1 C2 or else: ??=2 0.8846

10. Latent Class Analysis (LCA) Parameters Class membership probability Conditional item response probabilities ??=1|?=1 ; ??=2|?=1; ??=3|?=1 ; ??=1|?=2 ; ??=2|?=2; ??=3|?=2 ??=1|?=1+ ??=2|?=1+ ??=3|?=1 = 1 ??=1|?=2+ ??=2|?=2+ ??=3|?=2 = 1

11. Example of Data & Output ID Low Mood Lack Pleasure Sleep Problems Fatigue Latent class 0.1254 C1 or else: ??=1 101 1 1 2 1 0.8846 C2 or else: ??=2 102 3 3 2 3 Indicators Class 1 Class 2 103 1 2 1 1 Low Mood 1 (Never) 0.0235 0.8463 104 2 1 3 2 2 (Sometimes) 0.1838 0.1309 3 (Most times) 0.7927 0.0228 Lack Pleasure 1 (Never) 0.0173 0.9097 2 (Sometimes 0.2183 0.0881 ??=1|?=1 3 (Most times) 0.7644 0.0022

12. Example of Data & Output ID Low Mood Lack Pleasure Sleep Problems Fatigue Latent class ? 0.1254 C1 or else: ??=1 C2 or else: ??=2 101 1 1 2 1 ? 0.8846 102 3 3 2 3 Indicators Class 1 Class 2 103 1 2 1 1 Low Mood 1 (Never) 0.0235 0.8463 104 2 1 3 2 2 (Sometimes) 0.1838 0.1309 3 (Most times) 0.7927 0.0228 Lack Pleasure 1 (Never) 0.0173 0.9097 ??=1|?=1 2 (Sometimes 0.2183 0.0881 3 (Most times) 0.7644 0.0022 ??=2|?=1 ??=3|?=1

13. Example of Data & Output ID Low Mood Lack Pleasure Sleep Problems Fatigue Latent class ? 0.1254 C1 or else: ??=1 C2 or else: ??=2 101 1 1 2 1 ? 0.8846 102 3 3 2 3 Indicators Class 1 Class 2 103 1 2 1 1 Low Mood 1 (Never) 0.0235 0.8463 104 2 1 3 2 2 (Sometimes) 0.1838 0.1309 3 (Most times) 0.7927 0.0228 Lack Pleasure 1 (Never) 0.0173 0.9097 ??=1|?=1 2 (Sometimes 0.2183 0.0881 3 (Most times) 0.7644 0.0022 ??=2|?=1 ??=1|?=1 + ??=2|?=1 + ??=3|?=1 = 1.0 ??=3|?=1

14. Example of Data & Output ID Low Mood Lack Pleasure Sleep Problems Fatigue Latent class ? 0.1254 C1 or else: ??=1 C2 or else: ??=2 101 1 1 2 1 ? 0.8846 102 3 3 2 3 Indicators Class 1 Class 2 103 1 2 1 1 Low Mood 1 (Never) 0.0235 0.8463 104 2 1 3 2 2 (Sometimes) 0.1838 0.1309 3 (Most times) 0.7927 0.0228 Lack Pleasure 1 (Never) 0.0173 0.9097 2 (Sometimes 0.2183 0.0881 3 (Most times) 0.7644 0.0022

15. Visualising Conditional Probabilities Indicators Class 1 Class 2 Low Mood 1 (Never) 0.0235 0.8463 Probability of answer "Most of the time" by Latent Class and by Symtom 2 (Sometimes) 0.1838 0.1309 1 3 (Most times) 0.7927 0.0228 0.9 0.8 Lack Pleasure 1 (Never) 0.0173 0.9097 0.7 2 (Sometimes 0.2183 0.0881 0.6 0.5 3 (Most times) 0.7644 0.0022 0.4 0.3 Sleep Probs 1 (Never) 0.0087 0.7895 0.2 0.1 2 (Sometimes 0.1284 0.1974 0 Low Mood Lack Pleasure Sleep Problems 3 (Most times) 0.7846 0.0131 Class 1 Class 2

16. Visualising Conditional Probabilities Indicators Class 1 Class 2 Low Mood 1 (Never) 0.0235 0.8463 2 (Sometimes) 0.1838 0.1309 3 (Most times) 0.7927 0.0228 Lack Pleasure 1 (Never) 0.0173 0.9097 2 (Sometimes 0.2183 0.0881 3 (Most times) 0.7644 0.0022 Sleep Probs 1 (Never) 0.0087 0.7895 2 (Sometimes 0.1284 0.1974 3 (Most times) 0.7846 0.0131

17. Visualising Conditional Probabilities

18. Constraining Conditional Probabilities Conditional probabilities Class 1 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Low Mood Lack Pleasure Sleep Problems Never Sometimes Most times

19. Latent Class Analysis (LCA) Parameters Class membership probability Conditional item response probability Posterior probability Each individual s probability of being in latent class 1 (x = 1) and latent class 2 (x = 2) based on pattern of responses to A, B, C, and D.

20. Example of Data & Output ID p(c1) p(c2) ID Low Mood Lack Pleasure Sleep Problems Fatigue 101 0.0432 0.9568 101 1 1 2 1 102 0.9693 0.0307 102 3 3 2 3 103 0.0985 0.9015 103 1 2 1 1 104 0.4245 0.5755 104 2 1 3 2

21. Example of Data & Output ID p(c1) p(c2) ID Low Mood Lack Pleasure Sleep Problems Fatigue 101 0.0432 0.9568 101 1 1 2 1 102 0.9693 0.0307 102 3 3 2 3 103 0.0985 0.9015 103 1 2 1 1 104 0.4245 0.5755 104 2 1 3 2

22. Example of Data & Output ID p(c1) p(c2) ID Low Mood Lack Pleasure Sleep Problems Fatigue 101 0.0432 0.9568 101 1 1 2 1 102 0.9693 0.0307 102 3 3 2 3 103 0.0985 0.9015 103 1 2 1 1 104 0.4245 0.5755 104 2 1 3 2 Entropy: Measure of precision of latent class membership: Ranges from 0 (most uncertain) to 1 (certain membership)

23. Summary Probability-based model: Association between latent classes and indicators probabilistic. Latent class affiliations estimated with error. Model testing: Associations between indicators and underlying latent categorical variables can be constrained for hypothesis testing.