text:
Aberdeen (disambiguation)

Aberdeen is a city in Scotland.

Aberdeen may also refer to:

Africa

 * Aberdeen, Sierra Leone
 * Aberdeen, Eastern Cape, South Africa

Hong Kong

 * Aberdeen, Hong Kong, an area and town on southwest Hong Kong Island
 * Aberdeen Channel, a channel between Ap Lei Chau (Aberdeen Island) and Nam Long Shan on the Hong Kong Island in Hong Kong
 * Aberdeen Country Park, a country park in Hong Kong Island
 * Aberdeen floating village, at Aberdeen Harbour, containing approximately 600 junks, which house an estimated 6,000 people
 * Aberdeen Harbour, a harbour between Aberdeen, Hong Kong and Ap Lei Chau (Aberdeen Island)
 * Aberdeen Tunnel, a tunnel in Hong Kong Island
 * Aberdeen Tunnel Underground Laboratory, an underground particle physics laboratory in Hong Kong Island
 * Ap Lei Chau or Aberdeen Island, an island of Hong Kong
 * Aberdeen (Hong Kong constituency), a constituency of Southern District Council

India

 * Aberdeen Bazaar, a shopping centre in Port Blair, South Andaman Island

Sri Lanka

 * Aberdeen Falls, a waterfall in Sri Lanka

Australia

 * Aberdeen, New South Wales
 * Aberdeen, South Australia, one of the early townships that merged in 1940 to create the town of Burra
 * Aberdeen, Tasmania, a suburb of the City of Devonport

Caribbean

 * Aberdeen, Jamaica, a town in Saint Elizabeth, Jamaica

Europe

 * Aberdeen (Parliament of Scotland constituency)
 * Aberdeen (UK Parliament constituency) 1832-1885
 * Aberdeen Burghs (UK Parliament constituency) 1801-1832
 * Aberdeen Central (Scottish Parliament constituency)
 * Aberdeen Central (UK Parliament constituency)
 * Aberdeen Donside (Scottish Parliament constituency)
 * County of Aberdeen, a historic county of Scotland whose county town was Aberdeen
 * Old Aberdeen, a part of the city of Aberdeen in Scotland

Canada

 * Aberdeen, community in the township of Champlain, Prescott and Russell County, Ontario
 * Aberdeen, Abbotsford, a neighbourhood in the City of Abbotsford, British Columbia
 * Aberdeen Centre, a shopping mall in Richmond, British Columbia
 * Aberdeen, Grey County, Ontario
 * Aberdeen, Kamloops, an area in the City of Kamloops, British Columbia
 * Aberdeen Lake (Nunavut), a lake in Kivalliq Region, Nunavut, Canada
 * Aberdeen, Nova Scotia, part of the Municipality of Inverness County, Nova Scotia
 * Aberdeen Parish, New Brunswick
 * Rural Municipality of Aberdeen No. 373, Saskatchewan
 * Aberdeen, Saskatchewan
 * Aberdeen Bay, a bay between southern Baffin Island and north-eastern Hector Island in the Nunavut territory
 * Aberdeen Township, Quebec, until 1960 part of Sheen-Esher-Aberdeen-et-Malakoff, now part of Rapides-des-Joachims, Quebec
 * Aberdeen River, a tributary of rivière aux Castors Noirs in Mauricie, Québec
 * New Aberdeen, Nova Scotia

United States

 * Aberdeen, Arkansas
 * Aberdeen, Colorado
 * Aberdeen, Florida
 * Aberdeen, Georgia
 * Aberdeen, Idaho
 * Aberdeen, Ohio County, Indiana
 * Aberdeen, Porter County, Indiana
 * Aberdeen, Kentucky
 * Aberdeen, Maryland
 * Aberdeen Proving Ground, a United States Army facility located near Aberdeen, Maryland

 * Aberdeen, Massachusetts, a neighborhood of Brighton, Boston
 * Aberdeen, Mississippi
 * Aberdeen Lake (Mississippi), a lake in northeast Mississippi on the Tennessee-Tombigbee Waterway, close to Aberdeen, Mississippi

 * Aberdeen Township, New Jersey
 * Aberdeen, North Carolina
 * Aberdeen Historic District (Aberdeen, North Carolina)

 * Aberdeen, Ohio
 * Aberdeen, South Dakota
 * Aberdeen Historic District (Aberdeen, South Dakota)

 * Aberdeen, Texas
 * Aberdeen (Disputanta, Virginia)
 * Aberdeen Gardens (Hampton, Virginia)
 * Aberdeen, Washington
 * Aberdeen Gardens, Washington

 * Aberdeen, West Virginia

Arts and entertainment

 * Aberdeen (2000 film), a 2000 Norwegian-British film
 * Aberdeen (2014 film), a 2014 Hong Kong film
 * Aberdeen (band), an American rock band
 * Aberdeen (song), by Cage The Elephant

Companies

 * Abrdn, formerly Standard Life Aberdeen
 * Aberdeen Asset Management

Education

 * Aberdeen Business School, Robert Gordon University, Aberdeen, Scotland
 * Aberdeen College, Aberdeen, Scotland
 * Aberdeen Grammar School, Aberdeen, Scotland
 * Aberdeen Hall, a university-preparatory school in Kelowna, British Columbia, Canada
 * Aberdeen High School (disambiguation)
 * University of Aberdeen, Aberdeen, Scotland

Sports

 * Aberdeen F.C., a Scottish professional football team
 * Aberdeen F.C. Women
 * Aberdeen GSFP RFC, an amateur rugby union club in Aberdeen, Scotland
 * Aberdeen IronBirds, a Minor League Baseball team in Aberdeen, Maryland, U.S.
 * Aberdeen L.F.C., a women's football team affiliated with Aberdeen F.C.

Transportation

 * Aberdeen Airport (disambiguation)
 * Aberdeen station (disambiguation)
 * Aberdeen Line, a British shipping company founded in 1825
 * Aberdeen (ship), the name of several ships

title: Aberdeen (disambiguation)


text:
Algae

Algae (, ; : alga  ) are any of a large and diverse group of photosynthetic, eukaryotic organisms. The name is an informal term for a polyphyletic grouping that includes species from multiple distinct clades. Included organisms range from unicellular microalgae, such as Chlorella, Prototheca and the diatoms, to multicellular forms, such as the giant kelp, a large brown alga which may grow up to 50 m in length. Most are aquatic and lack many of the distinct cell and tissue types, such as stomata, xylem and phloem that are found in land plants. The largest and most complex marine algae are called seaweeds, while the most complex freshwater forms are the Charophyta, a division of green algae which includes, for example, Spirogyra and stoneworts. Algae that are carried by water are plankton, specifically phytoplankton.

Algae constitute a polyphyletic group since they do not include a common ancestor, and although their plastids seem to have a single origin, from cyanobacteria, they were acquired in different ways. Green algae are examples of algae that have primary chloroplasts derived from endosymbiotic cyanobacteria. Diatoms and brown algae are examples of algae with secondary chloroplasts derived from an endosymbiotic red alga. Algae exhibit a wide range of reproductive strategies, from simple asexual cell division to complex forms of sexual reproduction.

Algae lack the various structures that characterize land plants, such as the phyllids (leaf-like structures) of bryophytes, rhizoids of non-vascular plants, and the roots, leaves, and other organs found in tracheophytes (vascular plants). Most are phototrophic, although some are mixotrophic, deriving energy both from photosynthesis and uptake of organic carbon either by osmotrophy, myzotrophy, or phagotrophy. Some unicellular species of green algae, many golden algae, euglenids, dinoflagellates, and other algae have become heterotrophs (also called colorless or apochlorotic algae), sometimes parasitic, relying entirely on external energy sources and have limited or no photosynthetic apparatus. Some other heterotrophic organisms, such as the apicomplexans, are also derived from cells whose ancestors possessed plastids, but are not traditionally considered as algae. Algae have photosynthetic machinery ultimately derived from cyanobacteria that produce oxygen as a by-product of photosynthesis, unlike other photosynthetic bacteria such as purple and green sulfur bacteria. Fossilized filamentous algae from the Vindhya basin have been dated back to 1.6 to 1.7 billion years ago.

Because of the wide range of types of algae, they have increasing different industrial and traditional applications in human society. Traditional seaweed farming practices have existed for thousands of years and have strong traditions in East Asia food cultures. More modern algaculture applications extend the food traditions for other applications include cattle feed, using algae for bioremediation or pollution control, transforming sunlight into algae fuels or other chemicals used in industrial processes, and in medical and scientific applications. A 2020 review found that these applications of algae could play an important role in carbon sequestration in order to mitigate climate change while providing lucrative value-added products for global economies.

Etymology and study
The singular alga is the Latin word for 'seaweed' and retains that meaning in English. The etymology is obscure. Although some speculate that it is related to Latin algēre, 'be cold', no reason is known to associate seaweed with temperature. A more likely source is alliga, 'binding, entwining'.

The Ancient Greek word for 'seaweed' was φῦκος (phŷkos), which could mean either the seaweed (probably red algae) or a red dye derived from it. The Latinization, fūcus, meant primarily the cosmetic rouge. The etymology is uncertain, but a strong candidate has long been some word related to the Biblical פוך (pūk), 'paint' (if not that word itself), a cosmetic eye-shadow used by the ancient Egyptians and other inhabitants of the eastern Mediterranean. It could be any color: black, red, green, or blue.

The study of algae is most commonly called phycology ; the term algology is falling out of use.

Classifications
One definition of algae is that they "have chlorophyll as their primary photosynthetic pigment and lack a sterile covering of cells around their reproductive cells". On the other hand, the colorless Prototheca under Chlorophyta are all devoid of any chlorophyll. Although cyanobacteria are often referred to as "blue-green algae", most authorities exclude all prokaryotes, including cyanobacteria, from the definition of algae.

The algae contain chloroplasts that are similar in structure to cyanobacteria. Chloroplasts contain circular DNA like that in cyanobacteria and are interpreted as representing reduced endosymbiotic cyanobacteria. However, the exact origin of the chloroplasts is different among separate lineages of algae, reflecting their acquisition during different endosymbiotic events. The table below describes the composition of the three major groups of algae. Their lineage relationships are shown in the figure in the upper right. Many of these groups contain some members that are no longer photosynthetic. Some retain plastids, but not chloroplasts, while others have lost plastids entirely.

Phylogeny based on plastid not nucleocytoplasmic genealogy:

Linnaeus, in Species Plantarum (1753), the starting point for modern botanical nomenclature, recognized 14 genera of algae, of which only four are currently considered among algae. In Systema Naturae, Linnaeus described the genera Volvox and Corallina, and a species of Acetabularia (as Madrepora), among the animals.

In 1768, Samuel Gottlieb Gmelin (1744–1774) published the Historia Fucorum, the first work dedicated to marine algae and the first book on marine biology to use the then new binomial nomenclature of Linnaeus. It included elaborate illustrations of seaweed and marine algae on folded leaves.

W. H. Harvey (1811–1866) and Lamouroux (1813) were the first to divide macroscopic algae into four divisions based on their pigmentation. This is the first use of a biochemical criterion in plant systematics. Harvey's four divisions are: red algae (Rhodospermae), brown algae (Melanospermae), green algae (Chlorospermae), and Diatomaceae.

At this time, microscopic algae were discovered and reported by a different group of workers (e.g., O. F. Müller and Ehrenberg) studying the Infusoria (microscopic organisms). Unlike macroalgae, which were clearly viewed as plants, microalgae were frequently considered animals because they are often motile. Even the nonmotile (coccoid) microalgae were sometimes merely seen as stages of the lifecycle of plants, macroalgae, or animals.

Although used as a taxonomic category in some pre-Darwinian classifications, e.g., Linnaeus (1753), de Jussieu (1789), Lamouroux (1813), Harvey (1836), Horaninow (1843), Agassiz (1859),  Wilson & Cassin (1864), in further classifications, the "algae" are seen as an artificial, polyphyletic group.

Throughout the 20th century, most classifications treated the following groups as divisions or classes of algae: cyanophytes, rhodophytes, chrysophytes, xanthophytes, bacillariophytes, phaeophytes, pyrrhophytes (cryptophytes and dinophytes), euglenophytes, and chlorophytes. Later, many new groups were discovered (e.g., Bolidophyceae), and others were splintered from older groups: charophytes and glaucophytes (from chlorophytes), many heterokontophytes (e.g., synurophytes from chrysophytes, or eustigmatophytes from xanthophytes), haptophytes (from chrysophytes), and chlorarachniophytes (from xanthophytes).

With the abandonment of plant-animal dichotomous classification, most groups of algae (sometimes all) were included in Protista, later also abandoned in favour of Eukaryota. However, as a legacy of the older plant life scheme, some groups that were also treated as protozoans in the past still have duplicated classifications (see ambiregnal protists).

Some parasitic algae (e.g., the green algae Prototheca and Helicosporidium, parasites of metazoans, or Cephaleuros, parasites of plants) were originally classified as fungi, sporozoans, or protistans of incertae sedis, while others (e.g., the green algae Phyllosiphon and Rhodochytrium, parasites of plants, or the red algae Pterocladiophila and Gelidiocolax mammillatus, parasites of other red algae, or the dinoflagellates Oodinium, parasites of fish) had their relationship with algae conjectured early. In other cases, some groups were originally characterized as parasitic algae (e.g., Chlorochytrium), but later were seen as endophytic algae. Some filamentous bacteria (e.g., Beggiatoa) were originally seen as algae. Furthermore, groups like the apicomplexans are also parasites derived from ancestors that possessed plastids, but are not included in any group traditionally seen as algae.

Evolution
Algae are polyphyletic thus their origin cannot be traced back to single hypothetical common ancestor. It is thought that they came into existence when photosynthetic coccoid cyanobacteria got phagocytized by a unicellular heterotrophic eukaryote (a protist), giving rise to double-membranous primary plastids. Such symbiogenic events (primary symbiogenesis) are believed to have occurred more than 1.5 billion years ago during the Calymmian period, early in Boring Billion, but it is difficult to track the key events because of so much time gap. Primary symbiogenesis gave rise to three divisions of archaeplastids, namely the Viridiplantae (green algae and later plants), Rhodophyta (red algae) and Glaucophyta ("grey algae"), whose plastids further spread into other protist lineages through eukaryote-eukaryote predation, engulfments and subsequent endosymbioses (secondary and tertiary symbiogenesis). This process of serial cell "capture" and "enslavement" explains the diversity of photosynthetic eukaryotes.

Recent genomic and phylogenomic approaches have significantly clarified plastid genome evolution, the horizontal movement of endosymbiont genes to the "host" nuclear genome, and plastid spread throughout the eukaryotic tree of life.

Relationship to land plants
Fossils of isolated spores suggest land plants may have been around as long as 475 million years ago (mya) during the Late Cambrian/Early Ordovician period, from sessile shallow freshwater charophyte algae much like Chara, which likely got stranded ashore when riverine/lacustrine water levels dropped during dry seasons. These charophyte algae probably already developed filamentous thalli and holdfasts that superficially resembled plant stems and roots, and probably had an isomorphic alternation of generations. They perhaps evolved some 850 mya and might even be as early as 1 Gya during the late phase of the Boring Billion.

Morphology

A range of algal morphologies is exhibited, and convergence of features in unrelated groups is common. The only groups to exhibit three-dimensional multicellular thalli are the reds and browns, and some chlorophytes. Apical growth is constrained to subsets of these groups: the florideophyte reds, various browns, and the charophytes. The form of charophytes is quite different from those of reds and browns, because they have distinct nodes, separated by internode 'stems'; whorls of branches reminiscent of the horsetails occur at the nodes. Conceptacles are another polyphyletic trait; they appear in the coralline algae and the Hildenbrandiales, as well as the browns.

Most of the simpler algae are unicellular flagellates or amoeboids, but colonial and nonmotile forms have developed independently among several of the groups. Some of the more common organizational levels, more than one of which may occur in the lifecycle of a species, are
 * Colonial: small, regular groups of motile cells
 * Capsoid: individual non-motile cells embedded in mucilage
 * Coccoid: individual non-motile cells with cell walls
 * Palmelloid: nonmotile cells embedded in mucilage
 * Filamentous: a string of connected nonmotile cells, sometimes branching
 * Parenchymatous: cells forming a thallus with partial differentiation of tissues

In three lines, even higher levels of organization have been reached, with full tissue differentiation. These are the brown algae, —some of which may reach 50 m in length (kelps) —the red algae, and the green algae. The most complex forms are found among the charophyte algae (see Charales and Charophyta), in a lineage that eventually led to the higher land plants. The innovation that defines these nonalgal plants is the presence of female reproductive organs with protective cell layers that protect the zygote and developing embryo. Hence, the land plants are referred to as the Embryophytes.

Turfs
The term algal turf is commonly used but poorly defined. Algal turfs are thick, carpet-like beds of seaweed that retain sediment and compete with foundation species like corals and kelps, and they are usually less than 15 cm tall. Such a turf may consist of one or more species, and will generally cover an area in the order of a square metre or more. Some common characteristics are listed:
 * Algae that form aggregations that have been described as turfs include diatoms, cyanobacteria, chlorophytes, phaeophytes and rhodophytes. Turfs are often composed of numerous species at a wide range of spatial scales, but monospecific turfs are frequently reported.
 * Turfs can be morphologically highly variable over geographic scales and even within species on local scales and can be difficult to identify in terms of the constituent species.
 * Turfs have been defined as short algae, but this has been used to describe height ranges from less than 0.5 cm to more than 10 cm. In some regions, the descriptions approached heights which might be described as canopies (20 to 30 cm).

Physiology
Many algae, particularly species of the Characeae, have served as model experimental organisms to understand the mechanisms of the water permeability of membranes, osmoregulation, turgor regulation, salt tolerance, cytoplasmic streaming, and the generation of action potentials.

Phytohormones are found not only in higher plants, but in algae, too.

Symbiotic algae
Some species of algae form symbiotic relationships with other organisms. In these symbioses, the algae supply photosynthates (organic substances) to the host organism providing protection to the algal cells. The host organism derives some or all of its energy requirements from the algae. Examples are:

Lichens
Lichens are defined by the International Association for Lichenology to be "an association of a fungus and a photosynthetic symbiont resulting in a stable vegetative body having a specific structure". The fungi, or mycobionts, are mainly from the Ascomycota with a few from the Basidiomycota. In nature, they do not occur separate from lichens. It is unknown when they began to associate. One mycobiont associates with the same phycobiont species, rarely two, from the green algae, except that alternatively, the mycobiont may associate with a species of cyanobacteria (hence "photobiont" is the more accurate term). A photobiont may be associated with many different mycobionts or may live independently; accordingly, lichens are named and classified as fungal species. The association is termed a morphogenesis because the lichen has a form and capabilities not possessed by the symbiont species alone (they can be experimentally isolated). The photobiont possibly triggers otherwise latent genes in the mycobiont.

Trentepohlia is an example of a common green alga genus worldwide that can grow on its own or be lichenised. Lichen thus share some of the habitat and often similar appearance with specialized species of algae (aerophytes) growing on exposed surfaces such as tree trunks and rocks and sometimes discoloring them.

Coral reefs
Coral reefs are accumulated from the calcareous exoskeletons of marine invertebrates of the order Scleractinia (stony corals). These animals metabolize sugar and oxygen to obtain energy for their cell-building processes, including secretion of the exoskeleton, with water and carbon dioxide as byproducts. Dinoflagellates (algal protists) are often endosymbionts in the cells of the coral-forming marine invertebrates, where they accelerate host-cell metabolism by generating sugar and oxygen immediately available through photosynthesis using incident light and the carbon dioxide produced by the host. Reef-building stony corals (hermatypic corals) require endosymbiotic algae from the genus Symbiodinium to be in a healthy condition. The loss of Symbiodinium from the host is known as coral bleaching, a condition which leads to the deterioration of a reef.

Sea sponges
Endosymbiontic green algae live close to the surface of some sponges, for example, breadcrumb sponges (Halichondria panicea). The alga is thus protected from predators; the sponge is provided with oxygen and sugars which can account for 50 to 80% of sponge growth in some species.

Life cycle
Rhodophyta, Chlorophyta, and Heterokontophyta, the three main algal divisions, have life cycles which show considerable variation and complexity. In general, an asexual phase exists where the seaweed's cells are diploid, a sexual phase where the cells are haploid, followed by fusion of the male and female gametes. Asexual reproduction permits efficient population increases, but less variation is possible. Commonly, in sexual reproduction of unicellular and colonial algae, two specialized, sexually compatible, haploid gametes make physical contact and fuse to form a zygote. To ensure a successful mating, the development and release of gametes is highly synchronized and regulated; pheromones may play a key role in these processes. Sexual reproduction allows for more variation and provides the benefit of efficient recombinational repair of DNA damages during meiosis, a key stage of the sexual cycle. However, sexual reproduction is more costly than asexual reproduction. Meiosis has been shown to occur in many different species of algae.

Numbers
The Algal Collection of the US National Herbarium (located in the National Museum of Natural History) consists of approximately 320,500 dried specimens, which, although not exhaustive (no exhaustive collection exists), gives an idea of the order of magnitude of the number of algal species (that number remains unknown). Estimates vary widely. For example, according to one standard textbook, in the British Isles the UK Biodiversity Steering Group Report estimated there to be 20,000 algal species in the UK. Another checklist reports only about 5,000 species. Regarding the difference of about 15,000 species, the text concludes: "It will require many detailed field surveys before it is possible to provide a reliable estimate of the total number of species ..."

Regional and group estimates have been made, as well: and so on, but lacking any scientific basis or reliable sources, these numbers have no more credibility than the British ones mentioned above. Most estimates also omit microscopic algae, such as phytoplankton.
 * 5,000–5,500 species of red algae worldwide
 * "some 1,300 in Australian Seas"
 * 400 seaweed species for the western coastline of South Africa, and 212 species from the coast of KwaZulu-Natal. Some of these are duplicates, as the range extends across both coasts, and the total recorded is probably about 500 species. Most of these are listed in List of seaweeds of South Africa. These exclude phytoplankton and crustose corallines.
 * 669 marine species from California (US)
 * 642 in the check-list of Britain and Ireland

The most recent estimate suggests 72,500 algal species worldwide.

Distribution
The distribution of algal species has been fairly well studied since the founding of phytogeography in the mid-19th century. Algae spread mainly by the dispersal of spores analogously to the dispersal of cryptogamic plants by spores. Spores can be found in a variety of environments: fresh and marine waters, air, soil, and in or on other organisms. Whether a spore is to grow into an adult organism depends on the species and the environmental conditions where the spore lands.

The spores of freshwater algae are dispersed mainly by running water and wind, as well as by living carriers. However, not all bodies of water can carry all species of algae, as the chemical composition of certain water bodies limits the algae that can survive within them. Marine spores are often spread by ocean currents. Ocean water presents many vastly different habitats based on temperature and nutrient availability, resulting in phytogeographic zones, regions, and provinces.

To some degree, the distribution of algae is subject to floristic discontinuities caused by geographical features, such as Antarctica, long distances of ocean or general land masses. It is, therefore, possible to identify species occurring by locality, such as "Pacific algae" or "North Sea algae". When they occur out of their localities, hypothesizing a transport mechanism is usually possible, such as the hulls of ships. For example, Ulva reticulata and U. fasciata travelled from the mainland to Hawaii in this manner.

Mapping is possible for select species only: "there are many valid examples of confined distribution patterns." For example, Clathromorphum is an arctic genus and is not mapped far south of there. However, scientists regard the overall data as insufficient due to the "difficulties of undertaking such studies."

Ecology
Algae are prominent in bodies of water, common in terrestrial environments, and are found in unusual environments, such as on snow and ice. Seaweeds grow mostly in shallow marine waters, under 100 m deep; however, some such as Navicula pennata have been recorded to a depth of 360 m. A type of algae, Ancylonema nordenskioeldii, was found in Greenland in areas known as the 'Dark Zone', which caused an increase in the rate of melting ice sheet. Same algae was found in the Italian Alps, after pink ice appeared on parts of the Presena glacier.

The various sorts of algae play significant roles in aquatic ecology. Microscopic forms that live suspended in the water column (phytoplankton) provide the food base for most marine food chains. In very high densities (algal blooms), these algae may discolor the water and outcompete, poison, or asphyxiate other life forms.

Algae can be used as indicator organisms to monitor pollution in various aquatic systems. In many cases, algal metabolism is sensitive to various pollutants. Due to this, the species composition of algal populations may shift in the presence of chemical pollutants. To detect these changes, algae can be sampled from the environment and maintained in laboratories with relative ease.

On the basis of their habitat, algae can be categorized as: aquatic (planktonic, benthic, marine, freshwater, lentic, lotic), terrestrial, aerial (subaerial), lithophytic, halophytic (or euryhaline), psammon, thermophilic, cryophilic, epibiont (epiphytic, epizoic), endosymbiont (endophytic, endozoic), parasitic, calcifilic or lichenic (phycobiont).

Cultural associations
In classical Chinese, the word is used both for "algae" and (in the modest tradition of the imperial scholars) for "literary talent". The third island in Kunming Lake beside the Summer Palace in Beijing is known as the Zaojian Tang Dao (藻鑒堂島), which thus simultaneously means "Island of the Algae-Viewing Hall" and "Island of the Hall for Reflecting on Literary Talent".

Agar
Agar, a gelatinous substance derived from red algae, has a number of commercial uses. It is a good medium on which to grow bacteria and fungi, as most microorganisms cannot digest agar.

Alginates
Alginic acid, or alginate, is extracted from brown algae. Its uses range from gelling agents in food, to medical dressings. Alginic acid also has been used in the field of biotechnology as a biocompatible medium for cell encapsulation and cell immobilization. Molecular cuisine is also a user of the substance for its gelling properties, by which it becomes a delivery vehicle for flavours.

Between 100,000 and 170,000 wet tons of Macrocystis are harvested annually in New Mexico for alginate extraction and abalone feed.

Energy source
To be competitive and independent from fluctuating support from (local) policy on the long run, biofuels should equal or beat the cost level of fossil fuels. Here, algae-based fuels hold great promise, directly related to the potential to produce more biomass per unit area in a year than any other form of biomass. The break-even point for algae-based biofuels is estimated to occur by 2025.

Fertilizer
For centuries, seaweed has been used as a fertilizer; George Owen of Henllys writing in the 16th century referring to drift weed in South Wales: "This kind of ore they often gather and lay on great heapes, where it heteth and rotteth, and will have a strong and loathsome smell; when being so rotten they cast on the land, as they do their muck, and thereof springeth good corn, especially barley ... After spring-tydes or great rigs of the sea, they fetch it in sacks on horse backes, and carie the same three, four, or five miles, and cast it on the lande, which doth very much better the ground for corn and grass."

Today, algae are used by humans in many ways; for example, as fertilizers, soil conditioners, and livestock feed. Aquatic and microscopic species are cultured in clear tanks or ponds and are either harvested or used to treat effluents pumped through the ponds. Algaculture on a large scale is an important type of aquaculture in some places. Maerl is commonly used as a soil conditioner.

Nutrition
Naturally growing seaweeds are an important source of food, especially in Asia, leading some to label them as superfoods. They provide many vitamins including: A, B1, B2, B6, niacin, and C, and are rich in iodine, potassium, iron, magnesium, and calcium. In addition, commercially cultivated microalgae, including both algae and cyanobacteria, are marketed as nutritional supplements, such as spirulina, Chlorella and the vitamin-C supplement from Dunaliella, high in beta-carotene.

Algae are national foods of many nations: China consumes more than 70 species, including fat choy, a cyanobacterium considered a vegetable; Japan, over 20 species such as nori and aonori; Ireland, dulse; Chile, cochayuyo. Laver is used to make laverbread in Wales, where it is known as bara lawr. In Korea, green laver is used to make gim. It is also used along the west coast of North America from California to British Columbia, in Hawaii and by the Māori of New Zealand. Sea lettuce and badderlocks are salad ingredients in Scotland, Ireland, Greenland, and Iceland. Algae is being considered a potential solution for world hunger problem.

Two popular forms of algae are used in cuisine: Furthermore, it contains all nine of the essential amino acids the body does not produce on its own
 * Chlorella: This form of alga is found in freshwater and contains photosynthetic pigments in its chloroplast. It is high in iron, zinc, magnesium, vitamin B2 and Omega-3 fatty acids.
 * Spirulina: Known otherwise as a cyanobacterium (a prokaryote or a "blue-green alga")

The oils from some algae have high levels of unsaturated fatty acids. For example, Parietochloris incisa is high in arachidonic acid, where it reaches up to 47% of the triglyceride pool. Some varieties of algae favored by vegetarianism and veganism contain the long-chain, essential omega-3 fatty acids, docosahexaenoic acid (DHA) and eicosapentaenoic acid (EPA). Fish oil contains the omega-3 fatty acids, but the original source is algae (microalgae in particular), which are eaten by marine life such as copepods and are passed up the food chain. Algae have emerged in recent years as a popular source of omega-3 fatty acids for vegetarians who cannot get long-chain EPA and DHA from other vegetarian sources such as flaxseed oil, which only contains the short-chain alpha-linolenic acid (ALA).

Pollution control

 * Sewage can be treated with algae, reducing the use of large amounts of toxic chemicals that would otherwise be needed.
 * Algae can be used to capture fertilizers in runoff from farms. When subsequently harvested, the enriched algae can be used as fertilizer.
 * Aquaria and ponds can be filtered using algae, which absorb nutrients from the water in a device called an algae scrubber, also known as an algae turf scrubber.

Agricultural Research Service scientists found that 60–90% of nitrogen runoff and 70–100% of phosphorus runoff can be captured from manure effluents using a horizontal algae scrubber, also called an algal turf scrubber (ATS). Scientists developed the ATS, which consists of shallow, 100-foot raceways of nylon netting where algae colonies can form, and studied its efficacy for three years. They found that algae can readily be used to reduce the nutrient runoff from agricultural fields and increase the quality of water flowing into rivers, streams, and oceans. Researchers collected and dried the nutrient-rich algae from the ATS and studied its potential as an organic fertilizer. They found that cucumber and corn seedlings grew just as well using ATS organic fertilizer as they did with commercial fertilizers. Algae scrubbers, using bubbling upflow or vertical waterfall versions, are now also being used to filter aquaria and ponds.

Polymers
Various polymers can be created from algae, which can be especially useful in the creation of bioplastics. These include hybrid plastics, cellulose-based plastics, poly-lactic acid, and bio-polyethylene. Several companies have begun to produce algae polymers commercially, including for use in flip-flops and in surf boards.

Bioremediation
The alga Stichococcus bacillaris has been seen to colonize silicone resins used at archaeological sites; biodegrading the synthetic substance.

Pigments
The natural pigments (carotenoids and chlorophylls) produced by algae can be used as alternatives to chemical dyes and coloring agents. The presence of some individual algal pigments, together with specific pigment concentration ratios, are taxon-specific: analysis of their concentrations with various analytical methods, particularly high-performance liquid chromatography, can therefore offer deep insight into the taxonomic composition and relative abundance of natural algae populations in sea water samples.

Stabilizing substances
Carrageenan, from the red alga Chondrus crispus, is used as a stabilizer in milk products.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Algae


text:
Analysis of variance

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.

History
While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis testing in the 1770s. Around 1800, Laplace and Gauss developed the least-squares method for combining observations, which improved upon methods then used in astronomy and geodesy. It also initiated much study of the contributions to sums of squares. Laplace knew how to estimate a variance from a residual (rather than a total) sum of squares. By 1827, Laplace was using least squares methods to address ANOVA problems regarding measurements of atmospheric tides. Before 1800, astronomers had isolated observational errors resulting from reaction times (the "personal equation") and had developed methods of reducing the errors. The experimental methods used in the study of the personal equation were later accepted by the emerging field of psychology which developed strong (full factorial) experimental methods to which randomization and blinding were soon added. An eloquent non-mathematical explanation of the additive effects model was available in 1885.

Ronald Fisher introduced the term variance and proposed its formal analysis in a 1918 article on theoretical population genetics, The Correlation Between Relatives on the Supposition of Mendelian Inheritance. His first application of the analysis of variance to data analysis was published in 1921, Studies in Crop Variation I, This divided the variation of a time series into components representing annual causes and slow deterioration. Fisher's next piece, Studies in Crop Variation II, written with Winifred Mackenzie and published in 1923, studied the variation in yield across plots sown with different varieties and subjected to different fertiliser treatments. Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers.

Randomization models were developed by several researchers. The first was published in Polish by Jerzy Neyman in 1923.

Example
The analysis of variance can be used to describe otherwise complex relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is typically limited to dogs that are adult, pure-bred, and exemplary. A histogram of dog weights from a show might plausibly be rather complex, like the yellow-orange distribution shown in the illustrations. Suppose we wanted to predict the weight of a dog based on a certain set of characteristics of each dog. One way to do that is to explain the distribution of weights by dividing the dog population into groups based on those characteristics. A successful grouping will split dogs such that (a) each group has a low variance of dog weights (meaning the group is relatively homogeneous) and (b) the mean of each group is distinct (if two groups have the same mean, then it isn't reasonable to conclude that the groups are, in fact, separate in any meaningful way). In the illustrations to the right, groups are identified as X1, X2, etc. In the first illustration, the dogs are divided according to the product (interaction) of two binary groupings: young vs old, and short-haired vs long-haired (e.g., group 1 is young, short-haired dogs, group 2 is young, long-haired dogs, etc.). Since the distributions of dog weight within each of the groups (shown in blue) has a relatively large variance, and since the means are very similar across groups, grouping dogs by these characteristics does not produce an effective way to explain the variation in dog weights: knowing which group a dog is in doesn't allow us to predict its weight much better than simply knowing the dog is in a dog show. Thus, this grouping fails to explain the variation in the overall distribution (yellow-orange).

An attempt to explain the weight distribution by grouping dogs as pet vs working breed and less athletic vs more athletic would probably be somewhat more successful (fair fit). The heaviest show dogs are likely to be big, strong, working breeds, while breeds kept as pets tend to be smaller and thus lighter. As shown by the second illustration, the distributions have variances that are considerably smaller than in the first case, and the means are more distinguishable. However, the significant overlap of distributions, for example, means that we cannot distinguish X1 and X2 reliably. Grouping dogs according to a coin flip might produce distributions that look similar.

An attempt to explain weight by breed is likely to produce a very good fit. All Chihuahuas are light and all St Bernards are heavy. The difference in weights between Setters and Pointers does not justify separate breeds. The analysis of variance provides the formal tools to justify these intuitive judgments. A common use of the method is the analysis of experimental data or the development of models. The method has some advantages over correlation: not all of the data must be numeric and one result of the method is a judgment in the confidence in an explanatory relationship.

Classes of models
There are three classes of models used in the analysis of variance, and these are outlined here.

Fixed-effects models
The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.

Random-effects models
Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model.

Mixed-effects models
A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types.

Example
Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives.

Defining fixed and random effects has proven elusive, with multiple competing definitions.

Assumptions
The analysis of variance has been studied from several approaches, the most common of which uses a linear model that relates the response to the treatments and blocks. Note that the model is linear in parameters but may be nonlinear across factor levels. Interpretation is easy when data is balanced across factors but much deeper understanding is needed for unbalanced data.

Textbook analysis using a normal distribution
The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses:
 * Independence of observations – this is an assumption of the model that simplifies the statistical analysis.
 * Normality – the distributions of the residuals are normal.
 * Equality (or "homogeneity") of variances, called homoscedasticity—the variance of data in groups should be the same.

The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors ($$\varepsilon$$) are independent and $$\varepsilon \thicksim N(0, \sigma^2).$$

Randomization-based analysis
In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol. This randomization is objective and declared before the experiment is carried out. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher. This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University. Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox.

Unit-treatment additivity
In its simplest form, the assumption of unit-treatment additivity states that the observed response $$y_{i,j}$$ from experimental unit $$i$$ when receiving treatment $$j$$ can be written as the sum of the unit's response $$y_i$$ and the treatment-effect $$ t_j$$, that is $$y_{i,j}=y_i+t_j.$$ The assumption of unit-treatment additivity implies that, for every treatment $$j$$, the $$j$$th treatment has exactly the same effect $$t_j$$ on every experiment unit.

The assumption of unit treatment additivity usually cannot be directly falsified, according to Cox and Kempthorne. However, many consequences of treatment-unit additivity can be falsified. For a randomized experiment, the assumption of unit-treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit-treatment additivity is that the variance is constant.

The use of unit treatment additivity and randomization is similar to the design-based inference that is standard in finite-population survey sampling.

Derived linear model
Kempthorne uses the randomization-distribution and the assumption of unit treatment additivity to produce a derived linear model, very similar to the textbook model discussed previously. The test statistics of this derived linear model are closely approximated by the test statistics of an appropriate normal linear model, according to approximation theorems and simulation studies. However, there are differences. For example, the randomization-based analysis results in a small but (strictly) negative correlation between the observations. In the randomization-based analysis, there is no assumption of a normal distribution and certainly no assumption of independence. On the contrary, the observations are dependent!

The randomization-based analysis has the disadvantage that its exposition involves tedious algebra and extensive time. Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach. Few statisticians object to model-based analysis of balanced randomized experiments.

Statistical models for observational data
However, when applied to data from non-randomized experiments or observational studies, model-based analysis lacks the warrant of randomization. For observational data, the derivation of confidence intervals must use subjective models, as emphasized by Ronald Fisher and his followers. In practice, the estimates of treatment-effects from observational studies generally are often inconsistent. In practice, "statistical models" and observational data are useful for suggesting hypotheses that should be treated very cautiously by the public.

Summary of assumptions
The normal-model based ANOVA analysis assumes the independence, normality, and homogeneity of variances of the residuals. The randomization-based analysis assumes only the homogeneity of the variances of the residuals (as a consequence of unit-treatment additivity) and uses the randomization procedure of the experiment. Both these analyses require homoscedasticity, as an assumption for the normal-model analysis and as a consequence of randomization and additivity for the randomization-based analysis.

However, studies of processes that change variances rather than means (called dispersion effects) have been successfully conducted using ANOVA. There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest.

Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance. Also, a statistician may specify that logarithmic transforms be applied to the responses which are believed to follow a multiplicative model. According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

Characteristics
ANOVA is used in the analysis of comparative experiments, those in which only the difference in outcomes is of interest. The statistical significance of the experiment is determined by a ratio of two variances. This ratio is independent of several possible alterations to the experimental observations: Adding a constant to all observations does not alter significance. Multiplying all observations by a constant does not alter significance. So ANOVA statistical significance result is independent of constant bias and scaling errors as well as the units used in expressing observations. In the era of mechanical calculation it was common to subtract a constant from all observations (when equivalent to dropping leading digits) to simplify data entry. This is an example of data coding.

Algorithm
The calculations of ANOVA can be characterized as computing a number of means and variances, dividing two variances and comparing the ratio to a handbook value to determine statistical significance. Calculating a treatment effect is then trivial: "the effect of any treatment is estimated by taking the difference between the mean of the observations which receive the treatment and the general mean".

Partitioning of the sum of squares

ANOVA uses traditional standardized terminology. The definitional equation of sample variance is $s^2 = \frac{1}{n-1} \sum_i (y_i-\bar{y})^2$, where the divisor is called the degrees of freedom (DF), the summation is called the sum of squares (SS), the result is called the mean square (MS) and the squared terms are deviations from the sample mean. ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means.

The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model. For example, the model for a simplified ANOVA with one type of treatment at different levels.

$$SS_\text{Total} = SS_\text{Error} + SS_\text{Treatments}$$

The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.

$$DF_\text{Total} = DF_\text{Error} + DF_\text{Treatments}$$

The F-test
The F-test is used for comparing the factors of the total deviation. For example, in one-way, or single-factor ANOVA, statistical significance is tested for by comparing the F test statistic

$$F = \frac{\text{variance between treatments}}{\text{variance within treatments}}$$ $$F = \frac{MS_\text{Treatments}}{MS_\text{Error}} = {{SS_\text{Treatments} / (I-1)} \over {SS_\text{Error} / (n_T-I)}}$$

where MS is mean square, $$I$$ is the number of treatments and $$n_T$$ is the total number of cases

to the F-distribution with $$I - 1$$ being the numerator degrees of freedom and $$n_T - I$$ the denominator degrees of freedom. Using the F-distribution is a natural candidate because the test statistic is the ratio of two scaled sums of squares each of which follows a scaled chi-squared distribution.

The expected value of F is $$1 + {n \sigma^2_\text{Treatment}} / {\sigma^2_\text{Error}}$$ (where $$n$$ is the treatment sample size) which is 1 for no treatment effect. As values of F increase above 1, the evidence is increasingly inconsistent with the null hypothesis. Two apparent experimental methods of increasing F are increasing the sample size and reducing the error variance by tight experimental controls.

There are two methods of concluding the ANOVA hypothesis test, both of which produce the same result: The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level). For example, to test the hypothesis that various medical treatments have exactly the same effect, the F-test's p-values closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced. Such permutation tests characterize tests with maximum power against all alternative hypotheses, as observed by Rosenbaum. The ANOVA F-test (of the null-hypothesis that all treatments have exactly the same effect) is recommended as a practical test, because of its robustness against many alternative distributions.
 * The textbook method is to compare the observed value of F with the critical value of F determined from tables. The critical value of F is a function of the degrees of freedom of the numerator and the denominator and the significance level (α). If F ≥ FCritical, the null hypothesis is rejected.
 * The computer method calculates the probability (p-value) of a value of F greater than or equal to the observed value. The null hypothesis is rejected if this probability is less than or equal to the significance level (α).

Extended algorithm
ANOVA consists of separable parts; partitioning sources of variance and hypothesis testing can be used individually. ANOVA is used to support other statistical tools. Regression is first used to fit more complex models to data, then ANOVA is used to compare models with the objective of selecting simple(r) models that adequately describe the data. "Such models could be fit without any reference to ANOVA, but ANOVA tools could then be used to make some sense of the fitted models, and to test hypotheses about batches of coefficients." "[W]e think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences ..."

For a single factor
The simplest experiment suitable for ANOVA analysis is the completely randomized experiment with a single factor. More complex experiments with a single factor involve constraints on randomization and include completely randomized blocks and Latin squares (and variants: Graeco-Latin squares, etc.). The more complex experiments share many of the complexities of multiple factors. A relatively complete discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available.

There are some alternatives to conventional one-way analysis of variance, e.g.: Welch's heteroscedastic F test, Welch's heteroscedastic F test with trimmed means and Winsorized variances, Brown-Forsythe test, Alexander-Govern test, James second order test and Kruskal-Wallis test, available in onewaytests R

It is useful to represent each data point in the following form, called a statistical model: $$Y_{ij} = \mu + \tau_j + \varepsilon_{ij}$$ where
 * i = 1, 2, 3, ..., R
 * j = 1, 2, 3, ..., C
 * μ = overall average (mean)
 * τj = differential effect (response) associated with the j level of X; this assumes that overall the values of τj add to zero (that is, $\sum_{j = 1}^C \tau_j = 0$ )
 * εij = noise or error associated with the particular ij data value

That is, we envision an additive model that says every data point can be represented by summing three quantities: the true mean, averaged over all factor levels being investigated, plus an incremental component associated with the particular column (factor level), plus a final component associated with everything else affecting that specific data value.

For multiple factors
ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed factorial. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases. Consequently, factorial designs are heavily used.

The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz). All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare. The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results.

Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects." Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958). Some interactions can be removed (by transformations) while others cannot.

A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of analytical trickery) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications.

Associated analysis
Some analysis is required in support of the design of the experiment while other analysis is performed after changes in the factors are formally found to produce statistically significant changes in the responses. Because experimentation is iterative, the results of one experiment alter plans for following experiments.

The number of experimental units
In the design of an experiment, the number of experimental units is planned to satisfy the goals of the experiment. Experimentation is often sequential.

Early experiments are often designed to provide mean-unbiased estimates of treatment effects and of experimental error. Later experiments are often designed to test a hypothesis that a treatment effect has an important magnitude; in this case, the number of experimental units is chosen so that the experiment is within budget and has adequate power, among other goals.

Reporting sample size analysis is generally required in psychology. "Provide information on sample size and the process that led to sample size decisions." The analysis, which is written in the experimental protocol before the experiment is conducted, is examined in grant applications and administrative review boards.

Besides the power analysis, there are less formal methods for selecting the number of experimental units. These include graphical methods based on limiting the probability of false negative errors, graphical methods based on an expected variation increase (above the residuals) and methods based on achieving a desired confidence interval.

Power analysis
Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level. Power analysis can assist in study design by determining what sample size would be required in order to have a reasonable chance of rejecting the null hypothesis when the alternative hypothesis is true.

Effect size
Several standardized measures of effect have been proposed for ANOVA to summarize the strength of the association between a predictor(s) and the dependent variable or the overall standardized difference of the complete model. Standardized effect-size estimates facilitate comparison of findings across studies and disciplines. However, while standardized effect sizes are commonly used in much of the professional literature, a non-standardized measure of effect size that has immediately "meaningful" units may be preferable for reporting purposes.

Model confirmation
Sometimes tests are conducted to determine whether the assumptions of ANOVA appear to be violated. Residuals are examined or analyzed to confirm homoscedasticity and gross normality. Residuals should have the appearance of (zero mean normal distribution) noise when plotted as a function of anything including time and modeled data values. Trends hint at interactions among factors or among observations.

Follow-up tests
A statistically significant effect in ANOVA is often followed by additional tests. This can be done in order to assess which groups are different from which other groups or to test various other focused hypotheses. Follow-up tests are often distinguished in terms of whether they are "planned" (a priori) or "post hoc." Planned tests are determined before looking at the data, and post hoc tests are conceived only after looking at the data (though the term "post hoc" is inconsistently used).

The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across groups A, B and C to the mean of group D). Comparisons can also look at tests of trend, such as linear and quadratic relationships, when the independent variable involves ordered levels. Often the follow-up tests incorporate a method of adjusting for the multiple comparisons problem.

Follow-up tests to identify which specific groups, variables, or factors have statistically different means include the Tukey's range test, and Duncan's new multiple range test. In turn, these tests are often followed with a Compact Letter Display (CLD) methodology in order to render the output of the mentioned tests more transparent to a non-statistician audience.

Study designs
There are several types of ANOVA. Many statisticians base ANOVA on the design of the experiment, especially on the protocol that specifies the random assignment of treatments to subjects; the protocol's description of the assignment mechanism should include a specification of the structure of the treatments and of any blocking. It is also common to apply ANOVA to observational data using an appropriate statistical model.

Some popular designs use the following types of ANOVA:
 * One-way ANOVA is used to test for differences among two or more independent groups (means), e.g. different levels of urea application in a crop, or different levels of antibiotic action on several different bacterial species, or different levels of effect of some medicine on groups of patients. However, should these groups not be independent, and there is an order in the groups (such as mild, moderate and severe disease), or in the dose of a drug (such as 5 mg/mL, 10 mg/mL, 20 mg/mL) given to the same group of patients, then a linear trend estimation should be used. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test. When there are only two means to compare, the t-test and the ANOVA F-test are equivalent; the relation between ANOVA and t is given by $0.05$.
 * Factorial ANOVA is used when there is more than one factor.
 * Repeated measures ANOVA is used when the same subjects are used for each factor (e.g., in a longitudinal study).
 * Multivariate analysis of variance (MANOVA) is used when there is more than one response variable.

Cautions
Balanced experiments (those with an equal sample size for each treatment) are relatively easy to interpret; unbalanced experiments offer more complexity. For single-factor (one-way) ANOVA, the adjustment for unbalanced data is easy, but the unbalanced analysis lacks both robustness and power. For more complex designs the lack of balance leads to further complications. "The orthogonality property of main effects and interactions present in balanced data does not carry over to the unbalanced case. This means that the usual analysis of variance techniques do not apply. Consequently, the analysis of unbalanced factorials is much more difficult than that for balanced designs." In the general case, "The analysis of variance can also be applied to unbalanced data, but then the sums of squares, mean squares, and F-ratios will depend on the order in which the sources of variation are considered."

ANOVA is (in part) a test of statistical significance. The American Psychological Association (and many other organisations) holds the view that simply reporting statistical significance is insufficient and that reporting confidence bounds is preferred.

Generalizations
ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model. All consider the observations to be the sum of a model (fit) and a residual (error) to be minimized.

The Kruskal-Wallis test and the Friedman test are nonparametric tests which do not rely on an assumption of normality.

Connection to linear regression
Below we make clear the connection between multi-way ANOVA and linear regression.

Linearly re-order the data so that $$k$$-th observation is associated with a response $$y_k$$ and factors $$Z_{k,b}$$ where $$b \in \{1,2,\ldots,B\}$$ denotes the different factors and $$B$$ is the total number of factors. In one-way ANOVA $$B=1$$ and in two-way ANOVA $$B = 2$$. Furthermore, we assume the $$b$$-th factor has $$I_b$$ levels, namely $$\{1,2,\ldots,I_b\}$$. Now, we can one-hot encode the factors into the $ \sum_{b=1}^B I_b$ dimensional vector $$v_k$$.

The one-hot encoding function $$g_b : \{1,2,\ldots,I_b\} \mapsto \{0,1\}^{I_b}$$ is defined such that the $$i$$-th entry of $$g_b(Z_{k,b})$$ is $$g_b(Z_{k,b})_i = \begin{cases} 1 & \text{if } i=Z_{k,b} \\ 0 & \text{otherwise} \end{cases}$$ The vector $$v_k$$ is the concatenation of all of the above vectors for all $$b$$. Thus, $$v_k = [g_1(Z_{k,1}), g_2(Z_{k,2}), \ldots, g_B(Z_{k,B})]$$. In order to obtain a fully general $$B$$-way interaction ANOVA we must also concatenate every additional interaction term in the vector $$v_k$$ and then add an intercept term. Let that vector be $$X_k$$.

With this notation in place, we now have the exact connection with linear regression. We simply regress response $$y_k$$ against the vector $$X_k$$. However, there is a concern about identifiability. In order to overcome such issues we assume that the sum of the parameters within each set of interactions is equal to zero. From here, one can use F-statistics or other methods to determine the relevance of the individual factors.

Example
We can consider the 2-way interaction example where we assume that the first factor has 2 levels and the second factor has 3 levels.

Define $$a_i = 1$$ if $$Z_{k,1}=i$$ and $$b_i = 1$$ if $$Z_{k,2} = i$$, i.e. $$a$$ is the one-hot encoding of the first factor and $$b$$ is the one-hot encoding of the second factor.

With that, $$ X_k = [a_1, a_2, b_1, b_2, b_3 ,a_1 \times b_1, a_1 \times b_2, a_1 \times b_3, a_2 \times b_1, a_2 \times b_2, a_2 \times b_3, 1] $$ where the last term is an intercept term. For a more concrete example suppose that $$\begin{align} Z_{k,1} & = 2 \\ Z_{k,2} & = 1 \end{align}$$ Then, $$X_k = [0,1,1,0,0,0,0,0,1,0,0,1]$$

title: Analysis of variance