This sketch is suggested for all studies looking for associations of bacteria to any thing else. I will illustrate a sample results based on a sample of 100 people with self-declared Autism against a reference population at the end. The need for a better standard came out of this post, Microbiome Statistics for Dummies. As a FYI, applying all of the methods below had over 200 finding with P < 0.01 from the test sample.

Caveat emptor: This is based on 16s samples using self-declaration for Autism. The purpose of this post is not to report on microbiome shifts for Autism, but to show different statistics that should be used, but tend to be ignored.

Criteria

Measures must have the ability to compute statistical significance. Measures should be multi-faceted and not mono-faceted (i.e. comparing average only). The following measures are proposed as a new standard:

• Odds Ratio based on the Median of the targeted population.
• Odds Ratio based on the Incidence of being seen between populations
• Odds Ratio based on the Median of the reference population.
• Difference of Median between target and general population
• Difference of Means between target and general population
• Difference of Skews between target and general population

Odds Ratio

To compute an odds ratio (OR) and assess its statistical significance, you

• (1) calculate the OR from a 2×2 table, then
• (2) compute a confidence interval (CI) and/or a p‑value
  • via chi‑square test, or Fisher’s exact test

So we have for the three odds ratio cited above:

• Median Odds Ratio When Detected
  • Numbers above/below the Median of the target/reference populations
    • For Reference Population
    • For Target Population
• Incidence Odds Ratio When Detected
  • Number where it was detected, Number where it was not detected
    • For Reference Population
    • For Target Population

Difference of Skews

While this is not normally done, given the very high skewness (20+) seen with bacteria, it is a worthwhile investigation when sample sizes are sufficient. A description of how to do this is below.

Difference of Means

Given the high skewness of most bacteria, this is a well known standard approach will typically underestimate the significance.

Real Data using 100 Autism People

Bacteria Incidence P < 0.01

This first table caused me to do a double take because Bifidobacterium is generally low in Autism and yet we have a table with the odds of having Autism being increased with certain Bifidobacterium. The key factor to remember is that Bifidobacterium is rarely reported with Autism — but when it is reported, the amounts tend to be higher; additionally, certain species tend to be a lot more common with Autism.

To verify my computations, I also included the percentage of time it is seen with Autism and without. These Bifidobacterium are far more common to appear in samples.

tax_name	rank	Incidence Odds RATIO	Chi2	Symptom % Seen	Reference % Seen
Bifidobacterium catenulatum subsp. kashiwanohense	subspecies	2.8	39.80888	54	19
Bifidobacterium angulatum	species	2.8	32.81414	45	16
Hungateiclostridium	genus	2.4	17.91463	30	12
Hungateiclostridiaceae	family	2.4	17.7539	30	12
Bifidobacterium scardovii	species	2.1	13.64735	34	16
Clostridium chartatabidum	species	1.9	14.42216	48	24
Bifidobacterium catenulatum PV20-2	strain	1.9	15.85311	63	33
Bifidobacterium catenulatum	species	1.9	15.01648	64	35
Parascardovia	genus	1.8	9.703312	35	19
Bifidobacterium cuniculi	species	1.8	9.141149	34	18
Bifidobacterium gallicum	species	1.7	12.55468	82	48
ant endosymbionts	clade	1.7	7.530999	39	23
Candidatus Blochmanniella	genus	1.7	7.530999	39	23
Enterobacteriaceae incertae sedis	no rank	1.6	6.782294	39	24
Enterobacter hormaechei	species	1.6	6.962239	41	25
Moorella group	norank	1.6	8.616342	66	42
unclassified Bacteroidetes Order II.	order	1.6	8.740627	75	48
Bifidobacterium indicum	species	1.6	8.410521	76	49

Using Symptom Median P < 0.01

We have 152 bacteria identified. Bifidobacterium species again featured. For brevity, I show only the top 25 below. Remember using the Medium of those with Autism, 50% of Autistic people will be below, and 50% above — there is no need to show those counts.

Note that the top line is saying if the amount of Bifidobacterium is above 0.85 that the odds ratio is just .26 (greatly reduced). The Odds Ratio applies to those with the reported amount above the Median.

REMINDER: We are looking at only samples finding bacteria.

tax_name	Rank	Median	Odds Ratio	Chi2	Below	Above
Bifidobacterium	genus	0.8505	0.26	49.3	3114	817
Actinomycetota	phylum	1.0869	0.28	44.5	3126	872
Bifidobacteriaceae	family	0.764	0.28	43.4	3075	870
Bifidobacteriales	order	0.764	0.28	43.4	3075	870
Bifidobacterium catenulatum	species	0.01	0.29	39	1148	331
Bifidobacterium asteroides	species	0.005	0.29	36.8	872	255
Bifidobacterium subtile	species	0.0065	0.33	31.1	1093	357
Actinomycetes	class	0.71595	0.34	30.7	2978	1013
Bifidobacterium cuniculi	species	0.006	0.31	30.7	598	188
Parascardovia	genus	0.004	0.32	30.6	608	192
Leyella	genus	0.005	0.35	28.4	1314	454
Leyella stercorea	species	0.005	0.35	28.4	1314	454
Bacteroides rodentium	species	0.0695	2.72	26.1	1060	2878
Bifidobacterium gallicum	species	0.084	0.37	24.9	1508	559
Moraxellales	order	0.004	0.37	24.7	1597	595
Moraxellaceae	family	0.004	0.37	24.7	1597	595
Eukaryota	superkingdom	0.004	0.37	24.6	1089	401
Phocaeicola paurosaccharolyticus	species	0.025	2.62	24.1	1080	2828
Geopsychrobacteraceae	family	0.017	0.39	22.7	1400	540
Desulfuromusa	genus	0.017	0.39	22.7	1400	540
Burkholderiales genera incertae sedis	no rank	0.018	0.38	22.4	1024	393
Desulfuromonadales	order	0.015	0.39	22.4	1575	614
Desulfuromonadia	class	0.015	0.39	22.3	1574	614
Desulfuromonadaceae	family	0.016	0.39	21.9	1446	568
Psychrobacter	genus	0.003	0.39	21.5	1250	493

Using Reference Median P < 0.01

In this case, we have 89 bacteria. Bifidobacterium is very common again. To clarify matters a little: If the sample has Bifidobacterium reported and the amount is over 0.115, the odds of this person having Autism is 3.66.

The Odds Ratio applies to those with the reported amount above the Median

tax_name	Rank	Reference Median	Odds Ratio	Chi2	Symptom Below	Symptom Above
Bifidobacteriales	order	0.116	3.71	32.1	21	78
Bifidobacteriaceae	family	0.116	3.71	32.1	21	78
Bifidobacterium	genus	0.115	3.66	31.3	21	77
Actinomycetes	class	0.175	3.35	28.5	23	77
Caloramator indicus	species	0.005	0.09	25.8	34	3
Bifidobacterium gallicum	species	0.009	3.37	23.9	19	64
Actinomycetota	phylum	0.23	2.85	22.5	26	74
Hathewaya histolytica	species	0.159	0.39	18.3	71	28
Hathewaya	genus	0.159	0.39	18.3	71	28
Anaerovibrio lipolyticus	species	0.028	0.38	17.1	63	24
Rhodothermota	phylum	0.012	0.41	16.9	69	28
Rhodothermia	class	0.012	0.41	16.9	69	28
Rhodothermales	order	0.012	0.41	16.9	69	28
Clostridium thermosuccinogenes	species	0.008	0.37	16.3	57	21
Pseudoclostridium	genus	0.008	0.37	16.3	57	21
Anaerovibrio	genus	0.028	0.41	15.1	63	26
Enterobacteriaceae	family	0.055	2.12	12.7	32	68
Enterobacterales	order	0.057	2.12	12.7	32	68
Peptoniphilus	genus	0.052	0.46	12.6	65	30
Porphyromonas canis	species	0.005	0.39	12.1	46	18
Bifidobacterium adolescentis	species	0.011	2.18	12	28	61
Phocaeicola massiliensis	species	0.015	0.42	12	52	22
Olivibacter	genus	0.006	0.42	11.3	48	20
Phocaeicola paurosaccharolyticus	species	0.044	0.48	11.2	64	31
Eukaryota	superkingdom	0.002	3.5	11	8	28

Comparing Averages

A classic question on using averages: Do you include samples where a bacteria was not found as a zero, or exclude it from your average? I am inclined to suggest that both should be done.

tax_name	Rank	Symptom Average	Reference Average	Symptom Average With Zero	Reference Average With Zero
Bifidobacterium catenulatum	species	0.031	0.016	0.02	0.006
Bifidobacterium gallicum	species	0.845	0.293	0.695	0.142
Bifidobacterium angulatum	species	0.034	0.015	0.015	0.002
Bifidobacterium asteroides	species	0.007	0.005	0.003	0.001
Caloramator indicus	species	0.007	0.037	0.002	0.013
Bifidobacterium cuniculi	species	0.01	0.006	0.003	0.001
Bifidobacterium subtile	species	0.018	0.006	0.008	0.002
Bacteroides rodentium	species	0.187	0.397	0.182	0.366
Phocaeicola paurosaccharolyticus	species	0.035	0.06	0.033	0.055
Bifidobacterium indicum	species	0.028	0.019	0.021	0.009
Hathewaya histolytica	species	0.18	0.281	0.176	0.261
Bifidobacterium scardovii	species	0.005	0.009	0.002	0.001
Leyella stercorea	species	0.712	0.608	0.303	0.252
Phocaeicola sartorii	species	0.041	0.084	0.038	0.076
Anaerovibrio lipolyticus	species	0.055	0.114	0.047	0.098
Phascolarctobacterium succinatutens	species	0.033	0.061	0.022	0.045
Sarcina maxima	species	0.109	0.034	0.074	0.017
Clostridium thermosuccinogenes	species	0.008	0.015	0.006	0.012
Butyricimonas virosa	species	0.013	0.014	0.004	0.007
Bifidobacterium adolescentis	species	0.404	0.299	0.356	0.229
Veillonella montpellierensis	species	0.037	0.03	0.022	0.017
Sporolactobacillus putidus	species	0.039	0.018	0.015	0.005
Caloramator uzoniensis	species	0.005	0.01	0.002	0.005
Johnsonella ignava	species	0.065	0.051	0.063	0.047
Bacteroides cellulosilyticus	species	0.631	0.86	0.569	0.762

Averages to Median

Bacteria tend to have very high skewness rendering the use of means very unsafe. IMHO, medians is a far better statistics to use. As shown below, medians are almost always below the average because of extreme values. Often a mean will be at the 90%ile for a bacteria.

tax_name	Rank	Symptom Avarage	Reference Average	Symptom Median	Reference Median
Bifidobacterium catenulatum	species	0.031	0.016	0.01	0.003
Bifidobacterium gallicum	species	0.845	0.293	0.084	0.009
Bifidobacterium angulatum	species	0.034	0.015	0.009	0.004
Bifidobacterium asteroides	species	0.007	0.005	0.005	0.002
Caloramator indicus	species	0.007	0.037	0.002	0.005
Bifidobacterium cuniculi	species	0.01	0.006	0.006	0.003
Bifidobacterium subtile	species	0.018	0.006	0.007	0.003
Bacteroides rodentium	species	0.187	0.397	0.07	0.191
Phocaeicola paurosaccharolyticus	species	0.035	0.06	0.025	0.044
Bifidobacterium indicum	species	0.028	0.019	0.011	0.005
Hathewaya histolytica	species	0.18	0.281	0.092	0.159
Bifidobacterium scardovii	species	0.005	0.009	0.003	0.002
Leyella stercorea	species	0.712	0.608	0.005	0.003
Phocaeicola sartorii	species	0.041	0.084	0.017	0.033
Anaerovibrio lipolyticus	species	0.055	0.114	0.014	0.028
Phascolarctobacterium succinatutens	species	0.033	0.061	0.004	0.009
Sarcina maxima	species	0.109	0.034	0.017	0.007
Clostridium thermosuccinogenes	species	0.008	0.015	0.005	0.008
Butyricimonas virosa	species	0.013	0.014	0.011	0.006
Bifidobacterium adolescentis	species	0.404	0.299	0.044	0.011
Veillonella montpellierensis	species	0.037	0.03	0.019	0.007
Sporolactobacillus putidus	species	0.039	0.018	0.015	0.005
Caloramator uzoniensis	species	0.005	0.01	0.002	0.004
Johnsonella ignava	species	0.065	0.051	0.019	0.03
Bacteroides cellulosilyticus	species	0.631	0.86	0.02	0.079

Bottom Line

The purpose of this post is to discourage people from under-using the available data. Patterns can be counterintuitive, for example: Bifidobacteriums is detected less often, but when detected the amount is higher.

My goal for doing this deep dive is to look at different Odds Ratios. Odds ratios allows accurate prediction of likely and developing conditions. A sweet side effect is that it also allows priories on each bacteria to be objectively computed with a goal of higher success with interventions.

I have more investigations to do, especially double checking computations and doing cross validation with existing samples.